Automatically Generated Year Review Post 2014


The WordPress.com stats helper monkeys prepared a 2014 annual report for this blog.

Here’s an excerpt:

The concert hall at the Sydney Opera House holds 2,700 people. This blog was viewed about 31,000 times in 2014. If it were a concert at Sydney Opera House, it would take about 11 sold-out performances for that many people to see it.

Click here to see the complete report.

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Circular Motion and Orbits – A2 Physics Revision


Centripetal Acceleration

  • In circular motion, an object moves along a perfectly circular path
  • Its speed remains the same
  • The direction it travels in is constantly changing
  • Its velocity is therefore constantly changing
  • It is constantly accelerating towards the centre of the circle
  • The direction it’s moving in is always at 90° to the direction to the centre
  • An object with a force or acceleration acting on it perpendicular to its direction of travel will undergo circular motion
  • The acceleration towards the centre is called the centripetal acceleration
  • If the centripetal force / acceleration is removed, the orbiting object continues in the direction it was travelling that instant – a tangent to the circle
  • The centripetal force can be provided by tension in a piece of string, or by gravity, for example
  • (I’m sure you’re aware that the centrifugal force doesn’t exist – you can feel it because of your momentum straining against the centripetal force, but to the outside observer, there’s nothing there)
  • No work is done, because Work = Force × Distance Moved in Direction of Force, and the distance moved in the direction of the force is zero, because the orbiting object doesn’t actually get any closer to the centre, so the force is multiplied by zero
  • No work done means no energy transferred – objects in circular motion do not use up energy (well, unless they collide with something such as a bunch of air molecules, but we’re ignoring air resistance – especially when in space)
  • Now that we’ve got that out of the way…

Set Calculators to Radians

  1. l = r θ
    Where l is the length of an arc (part of the circumference of a circle), r is the radius of the circle and θ is the angle of the sector the arc is from
    Think of the sector as a slice of pizza, with r the length of one of the straight, cut sides and l the length of the curved crust
    This equation ONLY WORKS IF θ IS IN RADIANS
  2. ω = Δθ/Δt = v/r
    ω is the angular velocity, which is the rate of change of angular displacement. It’s measured in radians per second so can be calculated by dividing the angle the object has orbited by the time it took
    ω is equivalent to the linear (proper) velocity, v, divided by the radius of the circle
  3. a = vω = rω2 = v2/r
    The angular acceleration, a (with the same symbol as regular acceleration) is the angular velocity multiplied by the linear velocity, for some reason
    Anyway, substituting Equation 2 into it gives a = v2/r, which is useful
  4. F = mv2/r
    This comes from substituting the acceleration above into F = ma
    It’s an equation for the centripetal force in any circular motion
  5. v = (2πr) / T
    The linear velocity (at any time, in no particular direction… so the speed?) of the orbiting object is the distance it covers in one orbit divided by the time it takes to cover that distance, i.e. the circumference (2πr) divided by the period (T). This’ll be in metres per second if the circumference is in metres and period is in seconds
  6. F = -(GMm) / r2
    F is the gravitational force between two masses (although gravity isn’t a force – weight is), in Newtons, G is the gravitational constant, M is the mass of the larger object (at the centre of the circle), m is the mass of the smaller object (the one that’s orbiting the larger one) and r is the distance between the two objects
    This equation is just here so that I can number it in order to make a point later in the post. I may write a proper post about the gravitational field equations, although that’s looking a bit unlikely since the exam is… about 90 minutes from now, at the time of writing and I urgently need to do last-minute revision on relativistic and Doppler stuff…

Planets’ Orbits

  • This physics course treats planets’ orbital paths as perfect circles, unless otherwise specified
  • Planets orbiting the sun or satellites orbiting planets are modelled using circular motion
  • The planets / satellites themselves are treated as point masses – they have a centre of mass but their dimensions are ignored
  • Remember that the surface of a planet is still some distance away from where the point mass is – you may have to add the radius of the planet to the distance above the surface if you need the distance from the centre of the planet
  • Normally this won’t be necessary because you’ll be given the correct numbers in the question, but it could be used as an attempt to catch people out
  • Experimental Tip: To get a precise measurement of a planet’s orbital period, the time it takes to complete a lot of orbits can be measured and divided by the number of orbits (effectively averaging lots of repeats). Simply measuring the time for one orbit would mean a greater uncertainty
  • T2 = (4π2r3)/GM is known as Kepler’s Third Law
  • Since gravity provides the centripetal force when an object orbits another object in space, the gravitational force is equal to the centripetal force. You can equate the letter ‘F’s in equations 4 and 6 above, then substitute equation 5 in to eliminate v. If you rearrange the resulting expression, you should get Kepler’s Third Law

An Elliptical Orbit

  • An object in a circular orbit stays at the same distance from the thing it’s orbiting at all times, and its speed is constant
  • As mentioned earlier, no work is done by the centripetal force, so no energy is transferred
  • An object in an elliptical orbit, however, does get closer to and further away from the thing it’s orbiting, at different points in its orbit
  • This results in it speeding up and slowing down
  • The distance it moves in the direction of the centripetal force changes, so work is done and energy is transferred
  • The energy is converted between gravitational potential energy and kinetic energy
  • The total amount of energy does not change, so the kinetic energy added to the potential energy would produce the same result at any point in the orbit
  • By convention, gravitational potential energy is negative (as would be discussed in a post on non-uniform gravitational fields and potential wells, if I had time to write one)

Data, Formulae and Relationships Booklet

  • Equation 1 above is given (in the Mathematical Constants and Equations section at the bottom of the first page), as arc = rθ, although you’ll have to remember to USE RADIANS
  • It also has formulae for the area and circumference of a circle for a given radius
  • The formula booklet doesn’t bother with angular velocity – I’m not sure whether it’s part of the course or just used to explain equation 4
  • a = v2/r and F = mv2/r are in the formula booklet
  • Equation 5 doesn’t seem to be in the formula booklet, but it’s basically speed = distance divided by time (which you surely must remember!) with the circumference of a circle (given at the bottom of the first page of the booklet) as the distance
  • Equation 6 is in the formula booklet, in the Field and Potential section
  • Kepler’s Third Law doesn’t seem to be in the booklet, but I did explain how to derive it (and it only ever seems to be used in “show that” type questions)
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Simple Harmonic Motion (Part 3) – A2 Physics Revision


Have you read parts 1 and 2 of this post?

Spring Oscillator Period and Frequency

  • Since a = -4π2f2A sin(2πft) and s = A sin(2πft), you could say a = -4π2f2s
  • Equating Newton’s Second Law and Hooke’s Law, you get ma = -kx
  • Substituting in the expression for a above, cancelling the displacement with the extension of the spring (as they’re the same thing) and rearranging for F (which you probably won’t have to do, barring a nasty “show that” question), you get:
  • f = 1/(2π) √(m/k)
  • Or, since frequency and period are reciprocals:
  • T = 2π √(k/m)
  • These equations show how the period and frequency of a spring oscillator depend on the spring constant of the spring and the mass of the object
  • Remember: The frequency and period don’t change as the amplitude of the oscillations decreases – they’re independent of amplitude
  • The mass and spring constant affect the period of a spring oscillator

Pendulum Period and Frequency

  • Similarly, equating F = (mgs)/L with F = ma and substituting in a = -4π2f2s, you can obtain equations for the period and frequency of a pendulum:
  • f = 1/(2π) √(g/L)
  • T = 2π √(L/g)
  • The period and frequency of a pendulum are affected by the length of the string and the gravitational field strength / acceleration due to gravity
  • They’re not affected by the amplitude
  • For spring oscillators, the mass affects the period, but for pendulums it does not
  • Pendulums are used in old clocks because even when they lose energy due to air resistance and their amplitude decreases, their period is unaffected

Energy Conversions

  • Simple harmonic oscillators are faster in the centre than they are at the extremes of their oscillations (as shown by the velocity graph and the displacement graph)
  • Kinetic Energy is 1/2 mv2, so they have the most kinetic energy when they’re fastest and zero kinetic energy at the ends, where they stop for an instant
  • The kinetic energy is therefore in phase with the velocity, i.e. 90° degrees out of phase with displacement and 90° degrees out of phase with acceleration
  • Where does that kinetic energy go when it decreases to zero? It’s temporarily converted into another form of energy
  • In pendulums, it’s the gravitational potential energy they gain by swinging out to the side and increasing their height
  • In horizontal spring oscillators, it’s elastic potential energy from the compression or extension of the spring
  • In vertical spring oscillators, it’s a combination of elastic potential energy and gravitational potential energy
  • The total energy in the system (potential energy added to kinetic energy) is always the same
  • The potential energy is 90° out of phase with the kinetic energy
  • In case you need to actually add them up, Elastic Potential Energy is E = 1/2 kx2, Kinetic Energy is E = 1/2 mv2 and Gravitational Potential Energy is E = mgΔh
  • The total energy is the sum of those, or, for a spring oscillator, E = 1/2 kA2 where A is the amplitude and k is the spring constant

Energy Loss

  • Technically, the energy in the system does decrease, because the mass will usually be oscillating through something such as air
  • The mass collides with air particles and loses tiny amounts of kinetic energy. Eventually, this adds up
  • This is why the amplitude of pendulums gradually decreases – they are losing energy
  • The amplitude of the oscillator is proportional to the total energy in the system
  • The oscillator is said to be damped, as there is something taking energy from it
  • The damping effect would be greater if the oscillator were in a more dense medium, such as water
  • The energy decreases by a fixed percentage each oscillation
  • When its amplitude is greatest, a pendulum loses the most energy each swing, but when the pendulum is only swinging a tiny amount, it loses a much smaller amount of energy each swing
  • The rate of loss of energy is proportional to the amount of energy present
  • Does that sound familiar? The energy loss is exponential!
  • The loss of energy from damped oscillators follows exponential decay
  • Remember that the period and frequency are unaffected by the loss of energy / amplitude

Resonance

  • Similarly to how damping removes energy from an oscillating system, energy can be added to the system from outside
  • If the energy is applied at the right time, to coincide with the oscillations, the total energy in the system (and thus the amplitude) can be increased
  • Pushing a swing is the best analogy for this – you push when it’s going away from you, not when it’s coming towards you
  • Objects have a natural frequency which depends on various physical properties
  • If energy is applied to an object at its natural frequency, it oscillates and its amplitude increases dramatically
  • This can be a bad thing (bridges can be accidentally destroyed by marching on them, causing them to resonate), so damping is often used to counteract the effects of resonance
  • Like I said, I don’t have time to draw any more graphs, so you’ll have to imagine this next one (or just look it up in your notes / proper revision guide)
  • A graph with frequency on the x-axis and amplitude on the y-axis shows how the frequency external energy is applied to an object at affects the amplitude the object is made to oscillate at
  • The graph will have a sharp peak at the object’s natural frequency, and will curve towards zero on either side of the peak
  • If the object has been damped to reduce the effects of resonance, the peak will be flatter and wider (with a smaller maximum amplitude)

Data, Formulae and Relationships Booklet

  • T = 2π √(m/k) is in the formula booklet, so you can work out the period of a spring oscillator
  • If you want to derive it, F = kx is in the formula book (without the minus sign), there’s no way you could forget F = ma and a = -4π2f2s is in the formula booklet, disguised as a = -(2πf)2x, hidden in a larger expression at the start of the Oscillations section
  • f = 1/T is in the formula booklet, so you can work out the frequency from the period
  • The equations for a pendulum’s period and frequency are missing, as is the F = mgs/L approximation used to derive them, but I’m not sure whether you actually need those equations
  • The formula for the total energy in a spring oscillator is given and it’s also broken down into kinetic energy and elastic potential energy (but gravitational potential energy is ignored, so this must be a horizontal oscillator)
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Simple Harmonic Motion (Part 2) – A2 Physics Revision


Have you read part 1 of this post?

Without Calculus

  • It’s possible to figure out that velocity is cosine and acceleration is negative sine without actually differentiating (although you don’t get the actual equations)
  • To help explain, I’ve sketched you some graphs and scanned them in, like a professional taken photos of them with my phone
  • The first shows how the displacement varies. I’ve forgotten to actually label the axes, but I’ve annotated the peaks and some of the x-intercepts to explain the displacement, velocity and acceleration
A sinusoidal graph

Displacement and Time for Simple Harmonic Motion

  • At the first peak, displacement is at its maximum positive value
  • Here, the pendulum (for example) stops swinging the way it was going and starts to go back the other way, so the velocity at that instant is zero
  • The acceleration is at its maximum negative value, because the acceleration is proportional to the displacement but in the opposite direction
  • At the x-intercept in the middle of the graph, displacement is zero
  • The object is now up to its full speed, in the negative direction. (Pendulums are fastest in the middle)
  • The acceleration is zero, because it’s at the point of zero displacement
  • At the negative peak, the displacement is at its maximum negative value
  • Again, the velocity is zero because the oscillator has stopped and is about to go back the other way
  • The acceleration is fully positive, because the displacement is fully negative
  • At the x-intercept on the right of the graph, the displacement and acceleration are zero (like at the other x-intercept), but the velocity is positive
  • The oscillator is returning from negative displacement to enter positive displacement. (When a pendulum is in the middle, it’s sometimes swinging left to right and sometimes swinging right to left)

[Graph Cancelled]

  • I won’t draw the other two graphs (velocity and acceleration) because I don’t have time
  • (I have a lot more Physics to revise today)
  • I could have done it by drawing some empty axes and putting dots where the velocity would be at its maximum positive value, its maximum negative value and where it would cross the axes
  • These would be worked out by examining the peaks and x-intercepts on the displacement graph
  • I could then draw a curve through the points, knowing that it would form a phase-shifted sine curve, and I’d end up with a cosine graph of velocity and time
  • I could do the same thing for acceleration – work out where it will be at its greatest positive and negative values and where it will be zero, then draw a curve through it

Useful Points

  • No matter what’s in the brackets, a sine or a cosine always evaluates to something between -1 and 1 (inclusive)
  • If you’re asked for the maximum displacement, velocity or acceleration but not given all the information (such as what time it occurs), you can simplify your equation by assuming that the sine or cosine evaluates to 1 since that will produce a maximum
  • The same goes for the maximum displacement, velocity or acceleration in the negative direction – replace the sine or cosine with -1 (but the magnitude will be the same as the maximum in the positive direction anyway)
  • The gradient of the displacement graph is the velocity at that point and the gradient of the velocity graph is the displacement at that point
  • You can draw a tangent and measure the gradient instead of differentiating and substituting, but it’s less accurate
  • The graphs are steepest at the points where they cross the x-axis, so draw tangents there to obtain maximum gradients
  • The line from a peak to the next point where the line crosses the x-axis has the average gradient for that interval
  • The displacement could be drawn as a cosine curve. In this case, the behaviour is the same, but the timer has been started when displacement is maximum instead of when displacement is zero. The velocity would be a negative sine and the acceleration would be a negative cosine
  • The important thing is to get the phase shifts right. Think of the either rules of differentiation or what the gradient is at various points
  • Don’t forget radians mode!

Data, Formulae and Relationships Booklet

  • No formulae in this part of the post!

There’s also a part 3 of this post…

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Simple Harmonic Motion (Part 1) – A2 Physics Revision


Definition

  • Simple Harmonic Motion is a type of motion where:
  • An object oscillates back and forth about a point of zero displacement at the centre
  • The force on the object (and therefore the acceleration of the object) is proportional to the magnitude of the displacement
  • The force and acceleration are always towards the point of zero displacement (middle)
  • The period of the oscillation is the same regardless of the amplitude
  • The oscillation is sinusoidal

Springs Do It

  • Masses on springs oscillate with simple harmonic motion because:
  • The force provided by the spring is F = -kx
  • The magnitude of the force is proportional to the displacement
  • The spring constant, k, is the constant of proportionality
  • The force, F is in the opposite direction from the displacement / extension of the spring, x, hence the minus sign in the equation
  • (The force is what produces the acceleration on the mass)
  • If the mass is on a vertical spring, weight acts on it as well as the thrust or tension in the spring
  • The point of zero displacement would be where the weight balances the force from the spring, so that if the mass was moved in either direction the forces would no longer be balanced and there would be a resultant force back towards the centre
  • If the mass is on a horizontal spring, the point of zero displacement is where the spring is at its normal length
  • Moving the mass one way would compress the spring and moving it the other way would extend it, resulting in a force towards the centre
  • Because it’s simple harmonic motion, we know that the acceleration is proportional to the displacement (or extension, for a spring) but negative, i.e. a α -x
  • The formula booklet says that the constant of proportionality is k/m (this spring constant divided by the mass oscillating), so a = -(k/m)x

Pendulums Do It

  • Pendulums oscillate left and right with Simple Harmonic Motion
  • The restoring force (force that pulls the mass back to the centre) is provided by the horizontal component of the tension in the string
  • Whenever the mass is not directly under the point it’s suspended from, the direction of the tension is diagonal, so there’s a horizontal component
  • This component is directed towards the centre, and its size is proportional to the displacement
  • The force is approximately F = -(mgs)/L
  • F is the force, m is the mass of the object, g is the acceleration due to gravity / gravitational field strength and L is the length of the string
  • This equation is only true when the angle is small, because it ignores a bit of trigonometry

Where the Displacement and Time Equation Comes From (in Detail)

  • Because the oscillation is sinusoidal, the graph of displacement against time is a sine curve
  • The equation of the graph will be based on the form s = sin(θ), where θ is some angle
  • s is the symbol for displacement, which can be negative or positive depending on its direction
  • The sine of anything is always between -1 and 1, so unless the amplitude of the oscillator being modelled is exactly 1, it’ll need to be included in the equation:
  • s = A sin(θ), where A is the amplitude of the oscillator: the maximum distance between it and the centre
  • We want an equation for displacement, in terms of time, so θ isn’t much help
  • θ can be replaced with 2πft, which is the phase of an object oscillating with frequency f at time t
  • Notice the 2π – this equation is in radians. Make sure your calculator is in radians mode!
  • The equation becomes: s = A sin(2πft)
  • s is the displacement at time t, A is the amplitude, f is the frequency and t is the time

Getting Velocity and Acceleration Equations

  • Velocity is the rate of change of displacement (think of distance over time)
  • v = ds/dt, so differentiating s with respect to t produces an equation for v
  • s = A sin(2πft) is in radians, so it can be differentiated:
  • v = 2πfA cos(2πft)
  • Furthermore, acceleration is the rate of change of velocity
  • The equation for v can be differentiated again to get the equation for a
  • a = -4π2f2A sin(2πft)
  • It may help you to learn the rules for differentiating sines and cosines (If you study maths, you probably already know them)
  • Differentiating a sine makes it a cosine and differentiating a cosine produces a negative sine
  • The resulting sine or cosine is multiplied by the 2πf from inside the brackets, as 2πf is the coefficient of the t, which we’re differentiating with respect to
  • The equation for a has the 2πf from the v equation multiplied by the 2πf from the differentiation, to make 4π2f2

Comparing the Equations

  • s = A sin(2πft), v = 2πfA cos(2πft) and a = -4π2f2A sin(2πft)
  • The frequency of the displacement, velocity and acceleration is the same
  • There is a 90° phase difference between displacement and velocity
  • There is a further 90° phase difference between velocity and acceleration
  • The phase difference between displacement and acceleration is 180°
  • The acceleration is proportional to the displacement, but negative – this is because the force and acceleration are always towards the centre, i.e. in the opposite direction from displacement

Data, Formulae and Relationships Booklet

  • F = kx is in the booklet. Remember that F is in the opposite direction from x
  • You probably won’t need the approximation for the pendulum’s restoring force, which is good, because it’s not in the booklet
  • s = A sin(2πft) is in the formula booklet, but with the letter x instead of the letter s
  • x = A cos(2πft) is also in the formula booklet. This is used when the timer has been started with the oscillator at maximum positive displacement instead of when it was at zero displacement, causing it to form a cosine curve
  • The equations for v and a aren’t exactly in the formula booklet, because they differ depending on whether you’re using sine or cosine for displacement. What is in there, however, is:
  • d2x/dt2 = a = -(k/m)x = -(2πf)2x
  • (That’s telling you that a is the second derivative of displacement with respect to time, proportional to displacement by a constant of proportionality -k/x and equivalent to the displacement multiplied by 2πf twice and made negative)
  • The Simple Harmonic Motion equations given in the formula booklet are for spring oscillators and not pendulums

There are two more parts to this post…

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Capacitors (Part 3) – A2 Physics Revision


Have you read parts 1 and 2 of this post?

Exponential Discharging

  • Capacitors discharge quickly to begin with, but the discharge rate drops as they discharge
  • The charge stored by a capacitor decays exponentially when it is discharging
  • In an exponential relationship, the rate of change is proportional to the quantity present
  • The more electrons on the plate, the more electrons leave the plate per unit time
  • Since the number of electrons is hard to measure, the charge, Q, held by the capacitor is used in the graphs and equations

Equations for Discharging

  • Using the same format as for radioactive decay, but with electric charge instead of un-decayed nuclei, you get:
  • Q = Q0e-t/RC
  • Where RC is the time constant of the discharging circuit
  • From Ohm’s Law, V=IR you can see that V (potential difference) and I (current) are proportional when R (resistance) is constant
  • Because of this, the voltage across the capacitor also decays exponentially as it discharges:
  • V = V0e-t/RC
  • Finally, the current decays exponentially as well, because the current is the rate of flow of charge, and this rate decreases as the capacitor discharges (the charge flows more slowly)
  • I = I0e-t/RC

Exponential Charging Equations

  • Capacitors initially charge up quickly, but the charge rate drops as they charge (because it gets increasingly harder for the battery to force more electrons on to the same plate)
  • The current starts off large, but as the charge builds up, it starts to flow more slowly and the current decreases
  • The same goes for the charging current as the discharging current:
  • I = I0e-t/RC
  • But remember: The discharging current is in the opposite direction from the charging current, so one of them will have to be a negative current
  • The form of the charge and voltage curves are different, though. They start off increasing quickly and increase exponentially slower
  • It’s not y = e-x (starts off decreasing quickly and decreases exponentially slower)
  • …and it’s not y = ex (starts off increasing slowly and increases exponentially quicker)
  • It’s actually: Q = Q0(1-e-t/RC) and V = V0(1-e-t/RC)

In Summary

  • Charging isn’t just a regular exponential curve – it has the (1 – eto the power of stuff) bit
  • Discharging equations are slightly more simple
  • The equation for current is the same for both, BUT IN OPPOSITE DIRECTIONS
  • Here’s a table to make that clearer:
Charging Discharging
Charge, Q /Coulombs Q = Q0(1-e-t/RC) Q = Q0e-t/RC
Potential Difference, V /Volts  V = V0(1-e-t/RC) V = V0e-t/RC
Current, I /Amps  I = I0e-t/RC  I = I0e-t/RC (the other direction)

The Gradient

  • On these charging and discharging graphs of charge against time, the gradient is the change in charge divided by the change in time
  • dQ / dt… sound familiar?
  • Q/t = I = current
  • The gradient of a capacitor charging or discharging graph is the current

Data, Formulae and Relationships Booklet

  • Q = Q0e-t/RC appears to be in the booklet
  • The rest of the charging and discharging exponential equations don’t seem to be in there
  • Learn the blue ones in the table
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Capacitors (Part 2) – A2 Physics Revision


Holding More Electrons

  • Increasing the supply voltage (potential difference which charges the capacitor) means the capacitor can be charged up to a higher voltage, as more electrons can be crammed on to one plate
  • Making the plates closer together (which will involve making the dielectric thinner) will allow their fields to interact more strongly, holding more electrons to the plate
  • Making the plates larger (giving them a greater area) is another way to make the capacitor store more charge
  • Consider a collection of many different capacitors, all connected to the same battery voltage
  • The best capacitors will have thin, very insulating, dielectrics and large plates quite close together, as discussed above
  • They will be able to hold more electrons than other capacitors, even though they’re at the same voltage
  • The number of electrons which can be stored when at a certain voltage is set by the capacitance of the capacitor
  • If you read the labels on the best capacitors from the aforementioned selection, you’d find that they were the ones with the highest capacitances

Charge and Capacitance

  • To work out the capacitance of a capacitor, divide the charge it can store by the voltage it needs to charge it up to that charge
  • The symbol for charge is Q, and it’s measured in Coulombs (or C, for short)
  • Each electron has a charge of -1.6*10-19C, so the charge held by the capacitor is directly related to the number of electrons held (and an easier way of writing it, since the numbers aren’t so large)
  • The symbol for capacitance is C, and it’s measured in Farads (F, for short)
  • A capacitor with a capacitance of 1F could hold 1C of charge, if charged at a potential difference of 1V
  • That would actually make it massive by normal standards – most capacitors are measured in micro-Farads (μF), nano-Farads (nF) or even pico-Farads (pF)
  • The charge a capacitor can store is its capacitance multiplied by the charging potential difference. In other words:
  • Q = VC

Energy

  • If you plotted a graph with the charging potential difference on the x axis and the charge held by the capacitor on the y axis, it’d have a straight line
  • The gradient of the line would be the capacitance, and it would go through the origin
  • (Imagine Q = VC as y = mx + c, with Q as y and V as x)
  • Remember the definition of potential difference? Energy per unit charge
  • Multiplying potential difference by charge gives you the energy
  • On a graph of V and Q, this is the area under the line
  • Since it’s a straight line graph, this area is a triangle, with area (base * height) / 2
  • E = 1/2 QV
  • E is the energy the capacitor is storing; V is the potential difference; Q is the charge

Current and Stuff

  • As you know, current is the rate of flow of charge, Q/t
  • This means that charge is current multiplied by time
  • Q = I t
  • Q is charge (Coulombs), I is current (Amps) and t is time (seconds)
  • You now have a lot of simple equations which you can substitute into each other to eliminate unknowns
  • Substituting Q = VC into E = 1/2 VQ shows that E = 1/2 CV2
  • If you plotted potential difference against energy, you’d a parabola
  • The graph of E against V2 would be a straight line, with 1/2 C as its gradient
  • Don’t try to memorize all the possible graphs
  • If you come across a graph you’re not familiar with, have a look at what’s on the axes, and see whether there’s an equation linking those two variables. Once you find one, you may be able to work out what the gradient and the area represent

Use in Timing

  • Capacitors take different amounts of time to charge and discharge through different resistors
  • Because of this, circuits containing capacitors and resistors can be used to create time delays in electronic devices
  • Increasing the capacitance or the resistance makes the charging or discharging take longer
  • Different combinations of resistors and capacitors (‘RC Circuits’) can be compared by comparing their time constants
  • The time constant of an RC Circuit is RC. (The resistance of the resistor multiplied by the capacitance of the capacitor)
  • The symbol for the time constant is τ (the Greek letter Tau)
  • That means the equation is τ = RC
  • The units of the time constant work out as seconds

Time Constant

  • The time constant (RC) is also the time taken for the charge to fall to 0.37 times its original value
  • That’s 0.37Q0 if Q0 is the charge held by the capacitor before it starts to discharge
  • Another way of remembering it is Q0/e, as 0.37 = 1/e1
  • If charging was linear and not exponential, the time constant would be the amount of time the capacitor takes to fully discharge
  • In reality, the discharge rate decreases as the capacitor discharges (as explained in Part 1), so the capacitor only manages to discharge down to about 0.37Q0 in that time
  • The equation for a capacitor discharging is exponential (as will be described in the next post), so capacitors never ought to fully discharge, but in practise they do
  • (Just like how radioactive substances eventually defy their equation and fully decay)
  • It’s important that the time constant is in SECONDS (not minutes, hours, days or anything. Those are fine for a radioactive decay constant, but not for an RC circuit time constant!

Data, Formulae and Relationships Booklet

  • τ = RC is in the formula booklet
  • You’ll just have to remember that 0.37Q0 is reached after RC seconds
  • Q = VC  is in the formula booklet, written as C = Q/V
  • E = 1/2 QV is in the formula booklet (they’ve even added that E = 1/2 CV2)
  • It may help to remember Ohm’s Law (V = IR), which isn’t in there
  • Q = It is in the formula booklet, written as I = Δq/Δt
  • Combine to create more cool equations!
  • (Batteries not included. Equations sold separately)

There’s a third part to this post, as well!

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