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Wize University Physics Textbook (Master) > DC Circuits

Capacitors and Capacitance

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Capacitors and Capacitance


Capacitance is a measure of the ability to store charge, and thus electric energy, for later use.

Capacitors consist of two electrical conductors (one positively and one negatively charged) which are separated by an insulating material (dielectric).


The maximum charge that can be stored on the plates of the capacitor depends on the capacitance and the potential difference between the plates:

 Q=CV \boxed{ \ Q=CV \ }

  • CC is the capacitance, measured in Farads (F)
  • QQ is the charge on each plate
  • VV is the potential difference between the plates

Capacitors can have any shape. The capacitance depends on their geometry and the material between the two conductors.


Exam Tip
  • For capacitors connected to a battery, the voltage remains constant, but the charge will change if the capacitance changes.
  • For capacitors not connected to a battery, the charge remains constant, but the voltage will change if the capacitance changes.


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Parallel Plate Capacitor

This capacitor is made of parallel conducting plates. It stores the positive charges on one plate, and the negative ones on the other plate.


The capacitance of a parallel plate capacitor is given by:


 C=ε0Ad \boxed{ \ C=\dfrac{\varepsilon_0A}{d} \ }
  • AA is the area of the plate
  • dd is the separation between the plates
  • εo\varepsilon_o is the permittivity of free space, ε0=8.854×10−12\varepsilon_0=8.854\times10^{-12} F/m




The electric field inside an ideal capacitor (infinitely large metal plates) is uniform and constant:


 E=σε0=Qε0A \boxed{ \ E=\dfrac{\sigma}{\varepsilon_0}=\dfrac{Q}{\varepsilon_0A} \ }
  • EE is the electric field
  • σ\sigma is the surface charge density (charge per unit area)

The electric field outside an ideal capacitor is zero.




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Different Types of Capacitors




Planar Capacitors

The capacitance CC of a planar (parallel-plate) capacitor is:

 C=κ ε0Ad \boxed{ \ C=\kappa \ \dfrac{\varepsilon_0A}{d} \ }
  • AA is the area of the plate
  • dd is the separation between the plates
  • εo\varepsilon_o is the permittivity of free space, ε0=8.854×10−12\varepsilon_0=8.854\times10^{-12} F/m
  • κ\kappa is the dielectric constant of the material between the plates


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Cylindrical Capacitors

The capacitance CC of a cylindrical capacitor is:
 C=2πε0 Lln⁡(b/a) \boxed{ \ C=\dfrac{2\pi\varepsilon_0\ L}{\ln(b/a)} \ }

  • LL is the length
  • aa is the inner radius
  • bb is the outer radius












Spherical Capacitors

The capacitance CC of a spherical capacitor is:

 C=4πϵ0 abb−a \boxed { \ C = \dfrac{4\pi \epsilon_0\ ab}{b−a} \ }
  • aa is the inner radius
  • bb is the outer radius
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Example: Parallel-Plate Capacitor


A parallel-plate capacitor is connected to a battery and charged to Q1Q_1 on each plate. While connected, the distance between the plates is halved and the surface area is doubled. The new charge stored by the capacitor is Q2Q_2. What is the Q2/Q1Q_2/Q_1 ratio?

Let's write the equations we have for capacitance:

C=εoAdC=\dfrac{\varepsilon_oA}{d} and Q=CVQ=CV

Combining them we get the charge expressed as:

Q=εoAd VQ=\dfrac{\varepsilon_oA}{d}\ V

Let's write this equation for the charges before (1) and after (2). Because the capacitor is still connected to the battery, the voltage is the same before and after.


Before:

Q1=εoA1d1 VQ_1=\dfrac{\varepsilon_oA_1}{d_1}\ V


After:

Q2=εoA2d2 VQ_2=\dfrac{\varepsilon_o\bcs{A_2}}{\bct{d_2}}\ V

Now we have to replace distance d2=d12\bct{d_2=\dfrac{d_1}{2}} and area by A2=2A1\bcs{A_2=2A_1} :

Q2=2εo2A1d1 VQ_2=\dfrac{\bct2\varepsilon_o\bcs{2A_1}}{\bct{d_1}}\ V

=4 εoA1d1 V=4 \ \dfrac{\varepsilon_oA_1}{d_1}\ V

=4 Q1=4\ Q_1


Therefore the ratio of the charges is Q2Q1=4\dfrac{Q_2}{Q_1}=4


Practice: Charging a Capacitor with Another Capacitor


Consider a 99 V battery hooked up to a 9.009.00 nF capacitor. The battery charges this capacitor completely. Then, this capacitor is disconnected from the battery and connected to another (uncharged) 4.504.50 nF capacitor. What is the final charge on the new capacitor? Answer in nanoCoulombs.