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Capacitance

A capacitor is a device which stores electric charge and hence electrostatic energy. It consists of two conductors separated by an insulating medium. Depending upon the shape and size of the conductors and insulating medium, capacitors are available in varying shapes and sizes, but the basic configuration is two conductors carrying equal but opposite charges (Fig.).

Experiments show that the amount of charge Q stored in a capacitor is linearly proportional to the electric potential difference between the conductors, ΔV. Thus, we may write
Description: 36002.png
where C is a positive proportionality constant called capacitance. Physically, capacitance is a measure of the capacity of storing electric charge for a given potential difference ΔV.
 
Basic configuration of a capacitor
  1. Energy Stored in Capacitors
The energy stored in a capacitor is equal to the work done to charge it. Consider a capacitance C, holding a charge +q on one plate and -q on the other. Moving a small element of charge dq from one plate to the other against the potential difference V = q/C requires the work dW:
Description: 34226.png
 
where, W is the work, in joules
 
q is the charge, in coulombs
 
C is the capacitance, in farads
 
We can find the energy stored in a capacitance by integrating this equation. Starting with an uncharged capacitance (q = 0), the work done in moving a charge from one plate to the other until the plates have charges +Q and -Q respectively, is given as,
Description: 218885.png
Description: 218890.png
 
where, W is the energy in joules
 
C is the capacitance in farads
 
V is the voltage in volts

Parallel-Plate Capacitor

We consider two metallic plates of equal area A separated by a distance d, as shown in Fig. The charge on the top plate is +Q while the charge on the bottom plate is -Q, both distributed uniformly on the plates.
 
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(a) Parallel-plate capacitor
 
Description: 85993.png
(b) Electric field line for parallel-plate capacitor
 
Neglecting the edge effects and fringing effects and assuming an ideal situation, where field lines between the plates are straight lines, the electric field is calculated using Gauss’ law as,
 
Description: 33504.png
By choosing a Gaussian ‘pillbox’ with surface area A to enclose the charge on the positive plate (see Fig.), the electric field in the region between the plates is given as,
 
Description: 23874.png
Description: 210952.png
Gaussian surface for calculating the electric field between the plates
 
The potential difference between the plates is,
Description: 155990.png
 
From the definition of capacitance, we have,
Description: 30164.png

Cylindrical or Coaxial Capacitor

We consider a solid cylindrical conductor of radius a surrounded by a coaxial cylindrical shell of inner radius b, as shown in Fig. The length of both cylinders is L which is assumed to be much larger than (b - a), the separation of the cylinders, so that edge effects can be neglected. The capacitor is charged in such a way that the inner cylinder has charge +Q while the outer shell has a charge -Q. We want to calculate the capacitance of this cylindrical capacitor.
 
Description: 86036.png
(a) A cylindrical capacitor (b) End view of the capacitor. The electric field is nonvanishing only in the region a < r < b
 
To calculate the capacitance, we first compute the electric field both inside and outside the capacitor.
 
Due to the cylindrical symmetry of the system, we choose the Gaussian surface to be a coaxial cylinder with length l < L and radius r.
  1. Region r < a
The charge enclosed is zero, Qenc = 0, since any net charge in a conductor must reside on its surface. Therefore, the field is also zero.
Description: 36539.png
  1. Region a < r < b
Using Gauss’ law, we have,
Description: 24587.png
Description: 37986.png
where Description: 35584.png is the charge per unit length.
  1. Region r > b
The charge enclosed is zero, Qenc λl - λl = 0 since the Gaussian surface encloses equal but opposite charges from both conductors. Therefore, the field is also zero.
Description: 37986.png
It is seen that the electric field is nonvanishing only in the region a < r < b.
 
The potential difference is given by,
 
Description: 154802.png
Thus, the capacitance of the coaxial capacitor is given as,
 
Description: 38117.png
Once again, we see that the capacitance C depends only on the geometrical factors, La and b.

Spherical Capacitor

We consider a spherical capacitor which consists of two concentric spherical shells of radii a and b, as shown in Fig.The inner shell has a charge +Q and the outer shell has an equal but opposite charge -Q, both uniformly distributed over its surface. We want to calculate the capacitance of this spherical capacitor.
 
Description: 210962.png
(a) spherical capacitor with two concentric spherical shells of radii a and b (b) Gaussian surface for calculating the electric field
Similar to a cylindrical capacitor, the electric field is nonvanishing only in the region a < r < b. Using Gauss’ law, we obtain,
 
Description: 218924.png
Description: 154816.png
 
Therefore, the potential difference between the two conducting shells is,
 
Description: 86113.png
Thus, the capacitance of the spherical capacitor is given as,
 
Description: 32576.png
Again, the capacitance C depends only on the physical dimensions, a and b.





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