Electric Potential due to Charged non Conducting Sphere

 Electric Potential

Electric Potential

    The electric potential at a point is equal to the amount of work done by the external force in bringing the unit positive charge from infinity to that point inside the electric field without changing the kinetic energy. It is denoted by V. It is a scalar quantity.

Electric Potential

`V = \frac{W}{q_0}`

Here,

W = Work done 

`q_0 =` Unit Charge

The S.I. unit of electric potential is volt.

`\text{1 volt} = \frac {\text{1 Joule (J)}}{\text{1 Coulomb (C)}}`

    The electric potential at a point is 1 volt when the work done in bringing a 1 Coulomb charge from infinity to that point is 1 Joule.

Dimensional Formula of electric potential

`\text{Electric Potential} = \frac{\text{Work done}}{\text{Unit Charge}}`

`V = \frac{W}{q_0}`

`V = \frac{F S}{q_0}`        `(\because W = F S)`

`V = \frac{[M^1L^1T^{-2}][L^1]}{[T^1A^1]}`

`V = \frac{[M^1L^2T^{-2}]}{[T^1A^1]}`

`V = [M^1L^2T^{-3}A^{-1}]`

Question: Derive an expression for potential due to charged non-conducting spherical shell at following points (i) outer, (ii) at surface,(iii) inner point and also draw the graph for the variation of potential with the distance.

Electric Potential due to Charged non-Conducting Sphere

        Consider a non-conducting sphere of radius R is to be charged by charge of q. We have to calculate electric potential inside, at it's surface and outside the surface. Let the test point is at a distance of r from the center of the sphere.

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 Non-Conducting Sphere

    Consider a non-conducting sphere of radius R, centered O, and charged with q. The given charge to a non-conducting sphere is distributed throughout the entire volume 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 non-conducting sphere (r > R)

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

Electric Potential outside the non-conducting sphere r > R

`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^{-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 non-conducting sphere (r = R)

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

Electric Potential at the surface of a non-conducting sphere

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

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

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

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

Electric Potential at a point inside the non-conducting sphere

A given charge to a non-conducting sphere is uniformly distributed throughout its entire volume, which is why the intensity of the electric field is not uniform inside the sphere.

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

`V =  \frac{1}{4\pi\epsilon_0}\frac{q}{R} - \frac{1}{4\pi\epsilon_0 }\frac{q}{R^3}[  \frac{r^2}{2}  ]_R^r`

`V =  \frac{1}{4\pi\epsilon_0}\frac{q}{R} - \frac{1}{4\pi\epsilon_0 }\frac{q}{2R^3}[  r^2  ]_R^r`

`V =  \frac{1}{4\pi\epsilon_0}\frac{q}{R} - \frac{1}{4\pi\epsilon_0 }\frac{q}{2R^3}[  r^2 - R^2 ]`

`V =  \frac{1}{4\pi\epsilon_0}\frac{q}{R} - \frac{1}{4\pi\epsilon_0 }\frac{q r^2}{2R^3} + \frac{1}{4\pi\epsilon_0}\frac{q R^2}{2R^3}`

`V =  \frac{1}{4\pi\epsilon_0} q [ \frac{1}{R} - \frac{r^2}{2R^3} + \frac{ 1}{2R}]`

`V =  \frac{1}{4\pi\epsilon_0} q [ \frac{2R^2 - r^2 + R^2}{2R^3}]`

`V =  \frac{q}{4\pi\epsilon_0}  [ \frac{3R^2 - r^2 }{2R^3}]`

At centre r = 0

`V_{\text{centre}} =  \frac{q}{4\pi\epsilon_0}  [ \frac{3R^2}{2R^3}]`

`V_{\text{centre}} =  \frac{q}{4\pi\epsilon_0}  [ \frac{3}{2R}]`

`V_{\text{centre}} =  \frac{3}{2}  \frac{K q}{R}`

`V_{\text{centre}} =  \frac{3}{2}  V_{\text{surface}}`

Thus, the electric potential at the center of a charged non-conducting sphere is 1.5 times that on its surface.

The variation of electric potential with distance in a non-conducting sphere

The variation of electric potential with distance in a non-conducting sphere

Conclusion 

• Electric potential of a charged non-conducting sphere decreases with `r^2` from center to surface in a charged non-conducting sphere.
• Electric potential of a charged non-conducting sphere decreases with `r^{- 1}` from surface to infinity.
• Electric potential of a charged non-conducting sphere is maximum at the center.
• Electric potential of a charged non-conducting sphere is zero (minimum) at infinity.

Frequently Asked Questions

• What is a non-conducting sphere?
• 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 non-conducting sphere?
• How does the electric potential vary inside a non-conducting sphere?
• How does the electric potential change as you move closer to the center of a non-conducting sphere?
• How the electric potential of a non-conducting sphere relate to its charge distribution? Explane.
• Define electric potential difference.
• Define electric field.

Questions and Answer

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?

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

What is the electric potential of Earth Class 12?

Ans. The Earth's Electric potential is zero as a reference point in electrical calculations.

Is the electric potential of Earth zero?

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

What is zero potential?

Ans. Zero potential shows zero electric potential energy.

Why is electric potential zero at infinity?

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 non-conducting sphere?

Ans.    The electric potential distribution depends on the change and radius of the non-conducting sphere sphere.

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

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

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

Where,

K = Coulomb constant

Long Answer Type Question

    Find the expression for electric potential due to a non-conducting charged sphere at the outer point surface and inner point?