Chapter 2: Electrostatic Potential and Capacitance

In the above figure, we can see that the disc is divided into several charged rings. Let P be the point on the axis of the disc at a distance x from the centre of the disc.

The radius of the ring is r and the width is dr. dq is the charge on the ring, which is given as

dq = σdA = σ2πrdr

The potential is given as

The total potential at P is given as

Q/2πε₀R² (√R² + x² –x)

We know that the potential at point P is V = ∑V_i

Where V_i = q_i/4πε₀, r_i is the magnitude of the position vector P

V = 1/4 πε₀ ∑ q_i/r_pi

When the (x,y,z) plane is considered, the two charges lie on the z-axis and are separated by 2d. The potential is given as

Squaring the equation, we get

x² + y² + z² + [(q₁/q₂)²+1/(q₁/q₂)²-1] (2zd)+d²-0

Therefore, the equation of the sphere is

x² + y² + z² + 2ux + 2uy + 2wz + g = 0

The centre of the sphere is

And radius is

In the above figure, +q is the charge that got displaced from O towards (-d,0).

This is written as

At x = 0

Differentiating the equation with respect to x, we get

When x < 0, dU/dx > 0

And when x > 0, dU/dx < 0

Using this, we can define the charge on the particle to be F = -dU/dx

F = -dU/dx = 0

When

d²U/dx² = positive, equilibrium is stable

d²U/dx² = negative, equilibrium is unstable

d²U/dx² = 0, equilibrium is neutral

Therefore, when x = 0, d²U/dx² = (-2dq²/4πε0)(1/d⁶)(2d²) < 0

Which shows that the system is in unstable equilibrium.

Electric charges which are slow or stationary is called Electrostatic Charge, and the amount of work needed to move from one initial point to another is termed Electrostatic Potential. It can be measured in volts and is a scalar quantity. Capacitance, on the other hand, can be defined as the ability of a capacitor to store energy. They are measured in farads.