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Electricity and Magnetism
Electric Potential
Electric potential at a point in an electric field is defined as the work done per unit positive test charge in bringing it from infinity to that point without acceleration. It is a scalar quantity governing the electrostatic potential energy of charges.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Definition of electric potential, potential due to a point charge, superposition principle, relation between electric field and potential, equipotential surfaces, and potential energy of a system of point charges.
PGT extension
Potential due to continuous charge distributions (ring, disc, spherical shell, solid sphere), dipole potential at any general point, multipole expansion, Poisson's and Laplace's equations, and electrostatic potential energy of continuous charge configurations.
Prerequisites
Coulomb's Law and Electrostatic Force
Conservative nature of electrostatic field
Line integral of vector fields
Definition and Scalar Nature
Electric potential V at a point is given by $V = W/q_{0}$, where W is the work done by an external agent against electrostatic forces in bringing a positive test charge q0 from infinity.
It is a scalar quantity, meaning potentials due to multiple charges add algebraically.
The SI unit of electric potential is Joule per Coulomb (J/C) or Volt (V).
Relation between Electric Field and Potential
The electric field is the negative gradient of the electric potential: $E = -dV/dr$.
In three dimensions, $E = -((\partial V/\partial x)hat{i} + (\partial V/\partial y)hat{j} + (\partial V/\partial z)hat{k})$.
Electric field lines point in the direction of decreasing electric potential.
Equipotential Surfaces
An equipotential surface is a surface with a constant value of electric potential at all points on it.
The electric field vector is always normal (perpendicular) to the equipotential surface at every point.
No work is done in moving a test charge between two points on the same equipotential surface.
Electrostatic Potential Energy
The potential energy U of a system of charges is the total work done in assembling the charges from infinity to their current configuration.
For a two-point charge system, $U = (1/4piepsilon_{0}) * (q_{1}q_{2} / r)$.
Potential energy is associated with conservative forces and satisfies Delta $U = -W_{electric} = qDelta V$.
Derivations
Potential due to a Point Charge
Conditions: Static point charge q placed at the origin in vacuum
Consider a test charge q0 at a distance r from a source charge q.
The electrostatic force on the test charge at distance x is $F = (1/4piepsilon_{0}) * (qq_{0} / x^{2}) along the radial direction$.
The work done by the electric field in moving the test charge from r to infinity is $W = \int_{r}^{\infty} F dx$.
Two point charges q1 = +1 mu C and q2 = -2 mu C are placed at vertices (0, 0, 3) m and (0, 0, -3) m respectively. Find the electrostatic potential at point (4, 0, 0) m.
Determine distance from q1 (0,0,3) to (4,0,0): r1 = sqrt{(4-0)^2 + (0-0)^2 + (0-3)^2} = sqrt{16 + 9} = 5 m.
Determine distance from q2 (0,0,-3) to (4,0,0): r2 = sqrt{(4-0)^2 + (0-0)^2 + (0-(-3))^2} = sqrt{16 + 9} = 5 m.
Use superposition principle: $V = \frac{1}{4piepsilon_0} (\frac{q_1}{r_1} + \frac{q_2}{r_2})$.
Reference point for zero electric potential is assumed to be at infinity for localized charge distributions.
Electrostatic potential assumes static charges; time-varying magnetic fields introduce induced non-conservative electric fields where scalar potential alone is insufficient.
Common misconceptions
Common error: If the electric field at a point is zero, the electric potential must also be zero.
Correct: Electric potential can be non-zero even when the electric field is zero (e.g., inside a charged hollow conducting sphere).
Electric field is the negative spatial derivative of potential, so a constant non-zero potential yields a zero derivative (field).
Common error: Work done in moving a charge between two points depends on the path taken.
Correct: Work done by an electrostatic field is completely independent of the path taken.
Electrostatic force is conservative, meaning work depends only on the initial and final positions.
Quick revision
Electric potential is a scalar quantity obeying the principle of linear superposition.
Electric Potential of Point Charge: $V = \frac{1}{4piepsilon_0} \frac{q}{r}$
Electric Field from Potential: $E = -\frac{dV}{dr}$
Must remember
Electric potential $V$ is a scalar quantity defined as work done per unit positive test charge: $V = W/q_0$.
The electric field is the negative gradient of the electric potential: $\vec{E} = -\vec{\nabla} V$.
Electric field lines are always normal (perpendicular) to equipotential surfaces at every point.
No work is done in moving a test charge between two points located on the same equipotential surface.
The potential inside a charged conducting spherical shell is constant and equal to the potential at its surface: $V = \frac{1}{4\pi\epsilon_0}\frac{Q}{R}$.
Electrostatic force is conservative, meaning work done depends only on the initial and final positions, not on the path.
Glossary
Electric Potential ($V$)The work done by an external agent per unit positive test charge in bringing it from infinity to a point without acceleration.
Equipotential SurfaceA surface on which the electric potential is constant at every point.
Electrostatic Potential Energy ($U$)The total work done in assembling a system of charges from infinity to their current configuration.
Electric Dipole Moment ($p$)A vector quantity defined as the product of the magnitude of either charge and the distance between them, directed from negative to positive charge.
Electric Field vs Electric Potential
Parameter
Electric Field ($E$)
Electric Potential ($V$)
Nature
Vector quantity
Scalar quantity
SI Unit
Newton per Coulomb (N/C) or Volt per meter (V/m)
Joule per Coulomb (J/C) or Volt (V)
Spatial Relation
Points in the direction of decreasing potential
Scalar value whose negative gradient gives the field
Practice Quiz
Choose an option to reveal the explanation.
1. What is the SI unit of electric potential?
Explanation: Electric potential is defined as work done per unit charge ($V = W/q_0$), making its SI unit Joule per Coulomb (J/C), which is also known as Volt (V).
2. Which of the following statements about electric potential and electric field is correct?
Explanation: Inside a charged hollow conducting sphere, the electric field is zero, but the electric potential is constant and non-zero. $E = -dV/dr$, so zero field implies a constant potential.
3. What is the work done in moving a test charge $q_0$ along an equipotential surface between two points?
Explanation: Since the potential difference between any two points on an equipotential surface is zero, the work done $W = q_0 \Delta V$ is zero.
4. What is the relationship between the electric field $\vec{E}$ and electric potential $V$ in one dimension?
Explanation: The electric field is equal to the negative gradient of the electric potential, expressed in one dimension as $E = -\frac{dV}{dr}$.
5. What is the electric potential at a distance $r$ from a point charge $q$ in vacuum?
Explanation: The electric potential due to a point charge $q$ at a distance $r$ is given by $V = \frac{1}{4\pi\epsilon_0} \frac{q}{r}$.
6. How do potentials due to multiple point charges add up at a given point?
Explanation: Electric potential is a scalar quantity, so the total potential at a point due to a system of charges is the algebraic sum of the potentials due to individual charges.
7. What is the angle between the electric field vector and an equipotential surface at any point?
Explanation: The electric field vector is always normal (perpendicular, i.e., $90^\circ$) to the equipotential surface at every point.
8. What is the electrostatic potential energy $U$ of a two-point charge system with charges $q_1$ and $q_2$ separated by distance $r$?
Explanation: The electrostatic potential energy of a two-point charge system is $U = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r}$, assuming zero potential energy at infinity.
9. For a charged spherical shell of radius $R$ and total charge $Q$, what is the potential at a point inside the shell ($r < R$)?
Explanation: Inside a charged conducting spherical shell, the electric field is zero, and the potential remains constant and equal to its value at the surface, which is $\frac{1}{4\pi\epsilon_0} \frac{Q}{R}$.
10. Which of the following statements about electrostatic force and work is correct?
Explanation: Electrostatic force is a conservative force, meaning the work done by the field depends only on the initial and final positions and is completely independent of the path taken.
11. What is the electric potential at a distance $r$ along the axial line of an electric dipole with dipole moment $p$ (for $r \gg$ dipole length)?
Explanation: On the axial line of a dipole, $\theta = 0^\circ$, so the general formula $V = \frac{1}{4\pi\epsilon_0} \frac{p \cos\theta}{r^2}$ reduces to $V = \frac{1}{4\pi\epsilon_0} \frac{p}{r^2}$.
12. What is the electric potential on the equatorial plane of an electric dipole?
Explanation: On the equatorial plane of an electric dipole, $\theta = 90^\circ$, making $\cos\theta = 0$, so the electric potential is zero.
13. What is the potential energy $U$ of a point charge $q$ placed in an external electric potential $V(r)$?
Explanation: The potential energy of a point charge $q$ in an external potential $V(r)$ is given by $U = q V(r)$.
14. If the electric potential in a region is given by $V = 2x^2$ volts, what is the $x$-component of the electric field at $x = 1\text{ m}$?
Explanation: Using $E_x = -\frac{dV}{dx} = -\frac{d}{dx}(2x^2) = -4x$. At $x = 1\text{ m}$, $E_x = -4(1) = -4\text{ V/m}$.
15. Which of the following represents Laplace's equation in electrostatics?
Explanation: Laplace's equation applies to charge-free regions ($\rho = 0$) and is given by $\nabla^2 V = 0$.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. Two point charges $q_1 = +2\mu\text{C}$ and $q_2 = -2\mu\text{C}$ are placed $0.1\text{ m}$ apart. What is the electric potential at the midpoint of the line joining them?
Explanation: At the midpoint, distances from both charges are equal ($r = 0.05\text{ m}$). $V = \frac{1}{4\pi\epsilon_0} \left(\frac{q_1}{r} + \frac{q_2}{r}\right) = \frac{1}{4\pi\epsilon_0 r}(2\mu\text{C} - 2\mu\text{C}) = 0\text{ V}$ because potential is a scalar and the charges are equal and opposite.
2. The electric potential at a distance $r$ from a charged particle is $V$. What is the potential energy of a charge $q$ placed at that same point?
Explanation: The electrostatic potential energy $U$ of a charge $q$ in an electric potential $V$ is defined as $U = qV$.
3. If $10\text{ J}$ of work must be done to move an electric charge of $2\text{ C}$ from a point $A$ to point $B$, what is the potential difference $V_B - V_A$?
Explanation: Work done in moving a charge is $W = q(V_B - V_A)$. Therefore, $V_B - V_A = \frac{W}{q} = \frac{10\text{ J}}{2\text{ C}} = 5\text{ V}$.
4. What is the electric potential at a distance $r$ from the center of a thin spherical shell of radius $R$ carrying charge $Q$, when $r > R$?
Explanation: For points outside a charged spherical shell ($r > R$), the shell behaves as if all its charge is concentrated at its center, so $V = \frac{1}{4\pi\epsilon_0} \frac{Q}{r}$.
5. Four equal charges $q$ are placed at the four corners of a square of side $a$. What is the electric potential at the center of the square?
Explanation: The distance from each corner to the center is $r = \frac{a}{\sqrt{2}}$. The total potential is $V = 4 \times \frac{1}{4\pi\epsilon_0}\frac{q}{a/\sqrt{2}} = \frac{4\sqrt{2}q}{4\pi\epsilon_0 a}$.
6. In a certain region of space, electric potential is given by $V(x, y, z) = xy + yz + zx$. What is the $x$-component of the electric field at point $(1, 1, 1)$?
Explanation: The $x$-component of the electric field is $E_x = -\frac{\partial V}{\partial x} = -(y + z)$. At $(1, 1, 1)$, $E_x = -(1 + 1) = -2\text{ V/m}$.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. What does Poisson's equation relate electric potential to in electrostatics?
Explanation: Poisson's equation is $\nabla^2 V = -\frac{\rho}{\epsilon_0}$, which relates the second spatial derivative of electric potential to the volume charge density $\rho$.
2. If a unit positive charge is taken from one point to another over an equipotential surface, what is the work done?
Explanation: By definition, potential is constant on an equipotential surface, so $\Delta V = 0$ and work done $W = q \Delta V = 0$.
3. What is the electric potential due to an electric dipole at a distance $r$ along a line making an angle $\theta$ with the dipole axis?
Explanation: The potential due to a dipole at a general point $(r, \theta)$ for $r \gg$ dipole length is given by $V = \frac{1}{4\pi\epsilon_0} \frac{p \cos\theta}{r^2}$.
4. What is the electrostatic potential energy of a system of three point charges $q$, $2q$, and $3q$ placed at the vertices of an equilateral triangle of side $a$?
5. Which of the following is true regarding equipotential surfaces for a uniform electric field?
Explanation: For a uniform electric field, equipotential surfaces are a set of parallel planes perpendicular to the direction of the electric field lines.
6. If the electric potential $V$ is constant throughout a given region of space, what can be said about the electric field in that region?
Explanation: Since $\vec{E} = -\vec{\nabla} V$, if potential $V$ is constant, its gradient is zero, meaning the electric field is zero.
7. What is the dimension of electric potential in terms of fundamental quantities (Mass $M$, Length $L$, Time $T$, Current $A$)?
8. A charge $q$ is moved from infinity to a point in an electric field without acceleration. What is the work done by the external agent equal to?
Explanation: By work-energy theorem with zero change in kinetic energy, the work done by an external agent is equal to the change in potential energy of the system: $W_{ext} = \Delta U = qV$.
9. What is the potential at the surface of a uniformly charged solid sphere of radius $R$ and total charge $Q$?
Explanation: At the surface ($r = R$) of both a solid sphere and a spherical shell, the potential is given by $V = \frac{1}{4\pi\epsilon_0} \frac{Q}{R}$.
10. Why can two equipotential surfaces never intersect each other?
Explanation: If two equipotential surfaces intersected, the electric potential at the point of intersection would have two different values, which is physically impossible.