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

About Matt

I like writing, filmmaking, programming and gaming, and prefer creating media to consuming it. On the topic of consumption, I'm also a big fan of eating.
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