Electric potential is a crucial topic in Class 12 Physics (Electrostatics) for RBSE and CBSE board exams. It is defined as work done per unit charge (V = U/q). This chapter includes important derivations, formulas, and numericals, making it highly scoring and essential for board exam preparation and concept clarity.
Electric Potential
Definition: Electric potential in electrostatics is a fundamental concept in electromagnetism. It is defined as the work done per unit charge in an electric field.
Formula: `V = U/q`
Properties:
- Electric potential is a scalar quantity.
- S.I. unit of electric potential is the volt.
- `1 text{volt(V)} = frac{1 text{Joule (J)}}{1 text{Coulomb(C)}}`
- Dimensions of electric potential are `[M^1L^2T^{-3}A^{-1}]`
- Electric potential is independent of the test charge.
Derivation of Electric Potential from a Conducting Sphere
Consider a conducting sphere of radius R, centred at point O, and carrying a charge q. The charge given to a conducting sphere is distributed uniformly over the outer surface.
The relation between E and electric potential V is –
`V = – int_{infty}^r vec{E}.dvec{r}`
The observation point may lie in different regions where the electric potential is calculated.
Electric Potential Outside the Conducting Sphere (r > R)
`V = – int_{infty}^r vec{E}.dvec{r}`
 |
| Electric Potential due to Charged Conducting Sphere |
`V = – int_{infty}^r frac{1}{4piepsilon_0}frac{q}{r^2}hat{r}.dvec{r}`
`V = – frac{q}{4piepsilon_0} int_{infty}^r frac{1}{r^2} dr` `{because hat{r}.dvec{r}=dr}`
`V = – frac{q}{4piepsilon_0} int_{infty}^r r^{-2} dr`
`V = – frac{q}{4piepsilon_0} [ frac{r^{-2+1}}{-2+1} ]_{infty}^r`
`V = – frac{q}{4piepsilon_0} [ frac{r^{-1}}{-1} ]_{infty}^r`
`V = + frac{q}{4piepsilon_0} [ frac{1}{r} ]_{infty}^r`
`V = + frac{q}{4piepsilon_0} [ frac{1}{r}-frac{1}{infty} ]`
`V = + frac{q}{4piepsilon_0} [ frac{1}{r}-0 ]` `{because frac{1}{infty}=0}`
`V = + frac{q}{4piepsilon_0} [ frac{1}{r}]`
`V = + frac{1}{4piepsilon_0} frac{q}{r}`
`V = frac{Kq}{r}` `{because frac{1}{4piepsilon_0} = K}`
`V prop frac{1}{r}`
Thus, electric potential (V) increases as distance (r) decreases.
Electric Potential on the Surface of the Conducting Sphere (r = R)
At a point on the surface of the conducting sphere (r = R):
 |
| Electric Potential due to a Charged Conducting Sphere |
`V = frac{Kq}{r}`
`V = frac{Kq}{R}` `{because r=R}`
Thus, the electric potential has a constant value over the entire surface of the conducting sphere.
Electric Potential Inside Conducting Sphere (r < R)
 |
| Electric Potential due to a Charged Conducting Sphere |
A charge given to a conducting sphere is uniformly distributed over its outer surface. The electric field inside the sphere (r<R) is zero. For points outside the sphere (r>R), the electric field behaves as if the entire charge were concentrated at the center of the sphere.
The electric field inside a conducting sphere is zero everywhere.
The electric potential inside the conducting sphere is calculated in two parts:
(1.) From infinity to distance R (Surface), and
(2.) From distance R to r.
Thus, the integral is divided into two parts –
`V = [- int_infty^R vec{E}.dvec{r}]+[- int_R^rvec{E}.dvec{r}]`
`V = [ frac{1}{4piepsilon_0}frac{q}{R}]+[- int_R^r 0.dvec{r}]`
`V = frac{1}{4piepsilon_0}frac{q}{R} – 0`
`V = frac{1}{4piepsilon_0}frac{q}{R}`
`V = frac{K q}{R}` [Constant]
The electric potential inside a spherical shell is equal to the potential on its surface. This value is constant throughout the interior and is the maximum value of potential. The electric field inside the shell is zero, but the potential is not zero – it remains constant and equal to the surface potential.
Thus,
Electric potential inside the sphere equals the 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 and on the surface = `V = frac{K q}{R}`
(3.) Outside the sphere, the electric potential decreases as distance increases.
 |
| Change in Electric Potential with distance |
Conclusion
The electric potential inside a conducting sphere remains constant and equal to its surface value. Outside the sphere, the potential decreases inversely with distance (1/r). The potential is maximum inside and becomes zero 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.
The formula for 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. The dimensions of electric potential are `[M^1L^2T^{-3}A^{-1}]`.
Why is the electric potential of the Earth zero?
Electric potential is taken as 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 as zero as a reference point in electrical calculations.
Is the electric potential of Earth zero?
Ans. Yes, the electric potential of 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 the 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 to have 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.
What is the formula of outside electric potential due to a uniformly charged conducting sphere?
Ans. The formula for the 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.
Long Answer Type Question
Find the expression for electric potential due to a conducting charged sphere at the outer point surface and the inner point?
NCERT Chapter 2 Physics class 12
- Electric Potential
- Potential due to a Point Charge
- Electric Potential due to Charged Conducting Sphere with Derivation
- Electric Potential Due to Charged Conducting Sphere
- Electric Potential Due to Charged Non-Conducting Sphere
- Electric Potential due to Dipole at any Point
- Equipotential Surfaces and Properties of Equipotential Surface
- Electric Potential Due to Electric Dipole
- Electric Potential Due to a Group of Charges and Relation between Electric Field and Potential
- Electric Potential Energy of a System of Two-Point Charges
- Define the Electrostatic Potential Energy of a System of Two and Three-Point Charges
- Work in Rotation of Electric Dipole in Electric Field
- Potential Energy of a Dipole in an External Field
- Electrostatics of Conductors
- Dielectrics and polarization