Electric Potential due to Charged Conducting Sphere with Derivation

Electric Potential

Electric potential in electrostatics is a fundamental concept in electromagnetism, it is defined as the work done per unit charge in an electric field.

Electric potential is a scalar quantity.
S.I. unit of electric potential is volt.
`1 \text{volt(V)} = \frac{1 \text{Joule (J)}}{1 \text{Coulomb(C)}}`
Dimensions of electric potential is `[M^1L^2T^{-3}A^{-1}]`
Electric potential is independent of the test charge.

Derivation of Electric Potential from Conducting Sphere

Consider a conducting sphere with radius R, centered at point O, and carrying a charged q. The given charge to a conducting sphere is distributed throughout the entire outer surface of the sphere.

The relation of electric field E and electric potential V is -

`V = - \int_{\infty}^r \vec{E}.d\vec{r}`

The observation point may be in different conditions where the electric potential is to be calculated.

At a point outside the conducting sphere (r > R)

`V = - \int_{\infty}^r \vec{E}.d\vec{r}`

Electric Potential due to Charged Conducting Sphere

`V = - \int_{\infty}^r \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{r}.d\vec{r}`

`V = - \frac{q}{4\pi\epsilon_0} \int_{\infty}^r \frac{1}{r^2} dr` `{\because \hat{r}.d\vec{r}=dr}`

`V = - \frac{q}{4\pi\epsilon_0} \int_{\infty}^r r^{-2} dr`

`V = - \frac{q}{4\pi\epsilon_0} [ \frac{r^{-2+1}}{-2+1} ]_{\infty}^r`

`V = - \frac{q}{4\pi\epsilon_0} [ \frac{r^{-1}}{-1} ]_{\infty}^r`

`V = + \frac{q}{4\pi\epsilon_0} [ \frac{1}{r} ]_{infty}^r`

`V = + \frac{q}{4\pi\epsilon_0} [ \frac{1}{r}-\frac{1}{\infty} ]`

`V = + \frac{q}{4\pi\epsilon_0} [ \frac{1}{r}-0 ]` `{\because \frac{1}{\infty}=0}`

`V = + \frac{q}{4\pi\epsilon_0} [ \frac{1}{r}]`

`V = + \frac{1}{4\pi\epsilon_0} \frac{q}{r}`

`V = \frac{Kq}{r}` `{\because \frac{1}{4\pi\epsilon_0} = K}`

`V \prop \frac{1}{r}`

Thus, electric potential (V) increases as distance (r) decreases.

At the surface of the conducting sphere (r = R)

At a point on the surface of the conducting sphere, where r = R

Electric Potential due to Charged Conducting Sphere

`V = \frac{Kq}{r}`

`V = \frac{Kq}{R}` Constant `{\because r=R}`

At a point inside the conducting sphere (r < R)

At a point inside the conducting sphere, where r < R

Electric Potential due to Charged Conducting Sphere

A given charge to a conducting sphere is uniformly distributed throughout its entire outer surface, which is why the electric field inside the sphere is zero, and the electric field outside the sphere behaves as if the entire charge were concentrated at the center of the sphere.

In this condition, the electric field intensity is not uniform and dependent on the distance r.

The electric potential inside the charged conducting sphere can be calculated into two parts

(1.) From infinity to distance R (Surface), and

(2.) Fron distance R to r.

Thus, integral is divided into two parts -

`V = [- \int_\infty^R \vec{E}.d\vec{r}]+[- \int_R^r\vec{E}.d\vec{r}]`

`V = [ \frac{1}{4\pi\epsilon_0}\frac{q}{R}]+[- \int_R^r 0.d\vec{r}]`

`V = \frac{1}{4\pi\epsilon_0}\frac{q}{R} - \int_R^r\frac{1}{4\pi\epsilon_0}\frac{q}{R^3}r dr`

`V = \frac{1}{4\pi\epsilon_0}\frac{q}{R} - 0`

`V = \frac{1}{4\pi\epsilon_0}\frac{q}{R}`

`V = \frac{k q}{R}`

In this way, the electric potential inside the spherical shell is equal to that on the surface. This value is the maximum value for potential produced in a spherical shell.

Thus,

Electric potential inside = Electric potential on the surface

Change in Potential with distance in a charged spherical shell

(1.) The electric potential is constant and maximum inside the charged conducting spherical shell.

`V = \frac{k q}{R}`

(2.) Electric potential inside = Electric potential on the surface = `V = \frac{k q}{R}`

(3.) Outside the sphere the electric potential decreases with an increase in distance.

Change in Electric Potential with distance

Conclusion

The electric potential is constant and maximum inside the charged conducting spherical shell.

`V = \frac{k q}{R}`Electric potential inside = Electric potential on the surface = `V = \frac{k q}{R}
The electric potential of a charged conducting sphere decreases with `r^{- 1}` from surface to infinity.
The electric potential of a charged conducting sphere is maximum at the center, inside, and on the surface.
Electric potential of a charged conducting sphere is zero (minimum) at infinity.

Frequently asked Questions

What is a conducting sphere?
What is the difference between conducting and non-conducting spheres?
What is the electric potential at the surface of a conducting sphere?
How does the electric potential vary inside a conducting sphere?
How does the electric potential change as you move closer to the center of a conducting sphere?
Define electric potential difference.
Define electric field.

What is the electric potential of the earth?

Ans. In electrostatics, the electric potential of the Earth is taken to be zero.

What is the formula for electric potential?

Ans. Electric potential is defined as the electric potential energy per unit charge at a point in an electric field.

Formula of electric potential

`V = \frac{U}{q}`

Where,

U = Potential energy,

V = Potential and

q = Charge

What is the electric potential and its dimension?

Ans. In electrostatics, electric potential represents the energy per unit charge at a point in the electric field and Dimensions of electric potential is `[M^1L^2T^{-3}A^{-1}]`

Why is the electric potential of the earth zero?

Electric potential is taken zero as a reference point in electrical calculations.

What is the electric potential of Earth Class 12?

Ans. The electric potential of the Earth is taken to be zero as a reference point in electrical calculations.

Is the electric potential of Earth zero?

Ans. Yes, the electric potential of the Earth is considered zero as a reference point which is used in electrical calculations.

What is zero potential?

Ans. Zero potential shows zero electric potential energy.

Why is electric potential zero at infinity?

Ans. Electric potential decreases with an increase in distance from the charge, and at infinity, it becomes zero.

What is the charge of the Earth?

Ans. The Earth is approximately electrically neutral due to an equal number of positive and negative charges.

What is an example of a zero potential?

Ans. Earth's surface is considered as zero potential.

What factors influence the electric potential distribution around a conducting sphere?

Ans. The electric potential distribution depends on the charge and radius of the conducting sphere sphere.

What is the formula of outside electric potential due to a uniformly charged conducting sphere?

Ans. The formula of outside electric potential due to a uniformly charged conducting sphere is

`V = \frac{Kq}{r}`

Where,

K = Coulomb constant

How does the electric potential vary inside and outside a conducting sphere?

Ans.

(a) The electric potential inside a conducting sphere with uniform charge is constant.

(b) The electric potential outside a conducting sphere with uniform charge decreases with the inverse of the radial distance.

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Saturday 4 May 2024

Electric Potential due to Charged Conducting Sphere with Derivation