Electric Potential
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| 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.
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| 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 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 Spherical Shell
Consider a spherical shell 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.
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| Electric Potential due to Charged Spherical Shell |
At a point outside the spherical shell (r > R)
We know that electric potential
`V = - \int_{\infty}^r \vec E.d\vec r`
`V = - \int_{\infty}^r \frac{K q}{r^2}\hat r.d\vec r` `(\because \vec E = \frac{K q}{r^2}\hat r)`
`V = - \int_{\infty}^r \frac{K q}{r^2}dr` `(\because \hatr.d\vec r = dr)`
`V = - K q \int_{infty}^r \frac{1}{r^2} dr`
`V = - K q \int_{infty}^r r^{-2} dr`
`V = - K q [\frac{r^{- 2 + 1}}{- 2 + 1}]_{infty}^r`
`V = - K q [\frac{r^{-1}}{-1}]_{infty}^r`
`V = + K q [r^{-1}]_{infty}^r`
`V = + K q [\frac{1}{r}]_{infty}^r`
`V = + K q [\frac{1}{r} - \frac{1}{\infty}]`
`V = K q [\frac{1}{r} - 0]` `(\because \frac{1}{\infgy} = 0)`
`V = K q [\frac{1}{r}]`
`V_{out} = \frac{K q}{r}`
Where,
K = Coulomb Constant
`K = 9 \times 10^9 \frac{N m^2}{C^2}`
`K = \frac{1}{4 \pi \epsilon_0}`
q = Charge
So, we can say the electric potential is inversely proportional to the distance r from the center and it is zero at infinity.
At a point on the surface of spherical shell (r = R)
`V = - \int_{\infty}^R \vec E.d\vec r` `(\because r = R)` for surface.
So, we can find
`V_{\text{Surface}} = \frac{K q}{R}`
At a point inside the spherical shell (r < R)
For a point inside the spherical shell, the total work done can be divided in two part
`W = W_{\text{From infinity to R}} + W_{\text{R to r}}`
`V = V_{\text{From infinity to R}} + V_{\text{R to r}}`
Thus,
`V = - \int_{\infty}^r \vec E.d\vec r`
`V = - \int_{\infty}^R \vec E.d\vec r - \int_R^r \vec E_{\text{in}}.d\vec r`
`V = \frac{K q}{R} - \int_R^r \vec 0.d\vec r`
`V = \frac{K q}{R} + 0`
`V_{\text{in}} = \frac{K q}{R}`
Thus, 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.
Change in Potential with distance in a charged spherical shell
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 Non-Conducting Sphere
- Electric Potential due to Dipole at any Point
- Equipotential Surfaces and Properties of Equipotential Surface
- 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
- Capacitors and Capacitance, The Parallel Plate Capacitance and Effect of Dielectric on Capacitance
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