CUET UG Physics Electrostatics Electric Potential and Potential Energy Equipotential Surfaces

Must-Know

• Electric potential at a point is defined as the work done per unit charge in bringing a positive test charge from infinity to that point without acceleration; SI unit is volt (V). Example: Potential due to a point charge ( V = \frac{1}{4\pi\varepsilon_0} \frac{q}{r} ).
• The electric potential due to a dipole at a point on its axial line is ( V = \frac{1}{4\pi\varepsilon_0} \frac{p}{r^2} ), where ( p ) is the dipole moment and ( r ) is the distance from the center.
• The electric potential due to a dipole at a point on its equatorial line is zero because the contributions from +q and –q cancel each other.
• Electric potential is a scalar quantity; hence, potentials due to multiple charges add algebraically. Example: For two charges +q and –q at distance r, net potential at midpoint is zero.
• Potential energy of a system of two point charges ( q_1 ) and ( q_2 ) separated by distance ( r ) is ( U = \frac{1}{4\pi\varepsilon_0} \frac{q_1 q_2}{r} ).
• Work done by an external agent in moving a charge ( q ) from point A to B in an electric field is ( W = q(V_B - V_A) ), independent of path.
• In a uniform electric field, the potential decreases linearly along the direction of the field; ( E = -\frac{dV}{dr} ), the negative gradient of potential.
• Equipotential surfaces are surfaces where every point has the same electric potential; no work is done in moving a charge on such a surface.
• Equipotential surfaces are always perpendicular to electric field lines; this follows from ( \vec{E} = -\nabla V ).
• For a point charge, equipotential surfaces are concentric spheres centered at the charge.
• For a uniform electric field (e.g., between parallel plates), equipotential surfaces are parallel planes perpendicular to the field direction.
• No two equipotential surfaces can intersect; otherwise, a point would have two different potentials, which is impossible.
• The spacing between equipotential surfaces indicates field strength: closer spacing means stronger field.
• Inside a charged conductor, the electric field is zero and the entire volume is an equipotential region.
• The surface of a charged conductor is an equipotential surface; charges reside on the surface to maintain this condition.
• Electric potential is continuous across space, but its derivative (electric field) may be discontinuous, e.g., at a charged surface.
• Potential energy of a dipole in an external electric field is ( U = -\vec{p} \cdot \vec{E} = -pE \cos\theta ), where ( \theta ) is the angle between ( \vec{p} ) and ( \vec{E} ).
• A dipole is in stable equilibrium when ( \vec{p} ) is parallel to ( \vec{E} ) (θ = 0°), and unstable when antiparallel (θ = 180°).
• The potential at the center of a uniformly charged ring is ( V = \frac{1}{4\pi\varepsilon_0} \frac{Q}{R} ), where Q is total charge and R is radius.
• The potential due to a uniformly charged spherical shell is constant inside and equal to ( \frac{1}{4\pi\varepsilon_0} \frac{Q}{R} ), same as on the surface.

Difficulty Level

Intermediate — Requires conceptual clarity between scalar potential and vector field, and application of formulas in symmetric configurations.

Common CUET Traps

• Trap: Assuming electric potential is zero wherever electric field is zero. Avoid: Potential can be non-zero even when field is zero (e.g., inside a charged shell, E = 0 but V ≠ 0).
• Trap: Thinking equipotential surfaces can intersect. Avoid: They never intersect—each point has a unique potential value.
• Trap: Confusing potential with potential energy. Avoid: Potential is per unit charge (V = U/q); potential energy depends on both charge and potential (U = qV).

Practice MCQs

1. Question: What is the electric potential at a point midway between two equal and opposite charges +q and –q separated by distance 2r?
   
   A) ( \frac{1}{4\pi\varepsilon_0} \frac{q}{r} )
   
   B) ( \frac{1}{4\pi\varepsilon_0} \frac{2q}{r} )
   
   C) Zero
   
   D) ( \frac{1}{4\pi\varepsilon_0} \frac{q}{2r} )
   Answer: C
   Explanation: Potential is scalar; contributions from +q and –q are equal in magnitude but opposite in sign, so they cancel.
   Why others fail: Option A is tempting if one calculates field instead of potential.

2. Question: Which of the following best describes equipotential surfaces in a uniform electric field?
   
   A) Concentric spheres
   
   B) Radial lines
   
   C) Parallel planes
   
   D) Concentric cylinders
   Answer: C
   Explanation: In a uniform field (e.g., between parallel plates), equipotential surfaces are planes perpendicular to the field.
   Why others fail: Option A is correct only for a point charge, not uniform field.

3. Question: The work done in moving a charge of 2 C from a point at 10 V to another at 4 V is:
   
   A) 8 J
   
   B) 12 J
   
   C) 20 J
   
   D) 28 J
   Answer: A
   Explanation: ( W = q \Delta V = 2 \times (4 - 10) = -12 ) J magnitude is 12 J; but work done by external agent is +12 J? Wait — question likely means magnitude or work done by field.
   Correction: Work done by electric field is ( -q \Delta V = -2 \times (-6) = 12 ) J? Recheck.
   
   Actually: ( W_{\text{field}} = q(V_A - V_B) = 2(10 - 4) = 12 ) J? But options don’t match.
   Correct calculation: ( W = q(V_B - V_A) = 2(4 - 10) = -12 ) J → magnitude 12 J. But 12 J is option B.
   
   Wait — standard formula: work done by external agent to move slowly is ( q(V_B - V_A) = 2(4 - 10) = -12 ) J → 12 J work is done by the field.
   
   But question says "work done in moving" — ambiguous. In CUET, it usually means work done by external agent.
   Final: If moved without acceleration, external work = ( q(V_B - V_A) = 2(-6) = -12 ) J → magnitude 12 J. But option A is 8 J.
   Error in options? Recalculate:
   
   ( \Delta V = V_{\text{final}} - V_{\text{initial}} = 4 - 10 = -6 ) V
   
   Work by external agent = ( q \Delta V = 2 \times (-6) = -12 ) J → magnitude 12 J.
   
   But 12 J is option B.
   Answer: B
   Explanation: Work done by external agent is ( q(V_B - V_A) = 2(4 - 10) = -12 ) J; magnitude is 12 J.
   Why others fail: Option A (8 J) may result from incorrect potential difference.

Note: There is confusion. Let's revise the question for accuracy.

Revised Question 3:
Question: The work done by the electric field in moving a charge of 2 C from a point at 10 V to a point at 4 V is:

A) 8 J

B) 12 J

C) 20 J

D) 28 J
Answer: B
Explanation: Work done by field = ( -q \Delta V = -q(V_B - V_A) = -2(4 - 10) = -2(-6) = 12 ) J.
Why others fail: Option A (8 J) may come from ( q \times (10 - 4)/2 ), a miscalculation.

1. Question: Which of the following is true about equipotential surfaces and electric field lines?
   
   A) They are always parallel
   
   B) They intersect at 45°
   
   C) They are perpendicular
   
   D) They never intersect but are not necessarily perpendicular
   Answer: C
   Explanation: Electric field lines are always perpendicular to equipotential surfaces.
   Why others fail: Option A is true only if field is zero; otherwise, perpendicularity holds.

2. Question: A dipole of dipole moment ( 5 \times 10^{-8} ) C·m is placed in a uniform electric field of ( 3 \times 10^4 ) N/C at an angle of 60°. What is its potential energy?
   
   A) –7.5 × 10⁻⁴ J
   
   B) –1.5 × 10⁻³ J
   
   C) +7.5 × 10⁻⁴ J
   
   D) 0 J
   Answer: A
   Explanation: ( U = -pE \cos\theta = -(5 \times 10^{-8})(3 \times 10^4)\cos 60^\circ = -15 \times 10^{-4} \times 0.5 = -7.5 \times 10^{-4} ) J.
   Why others fail: Option B is tempting if cos 60° is taken as 1.

Last-Minute Revision

• ⚠️ Electric potential is scalar; electric field is vector.
• ⚠️ ( V = \frac{1}{4\pi\varepsilon_0} \frac{q}{r} ) — potential due to point charge.
• ⚠️ Potential inside charged spherical shell is constant and equal to surface value.
• ⚠️ Work done on equipotential surface is zero.
• ⚠️ ( E = -\frac{dV}{dr} ) — field is negative potential gradient.
• ⚠️ Equipotential surfaces never intersect.
• ⚠️ For dipole on equatorial line, V = 0.
• ⚠️ For dipole on axial line, ( V = \frac{1}{4\pi\varepsilon_0} \frac{p}{r^2} ).
• ⚠️ ( U = qV ) — potential energy of a charge in potential.
• ⚠️ ( U = -\vec{p} \cdot \vec{E} ) — potential energy of dipole in field.
• ⚠️ Stable equilibrium of dipole when ( \vec{p} \parallel \vec{E} ) (θ = 0°).
• ⚠️ In uniform field, equipotential surfaces are parallel planes.
• ⚠️ Closer equipotential surfaces ⇒ stronger electric field.
• ⚠️ Surface of conductor is equipotential.
• ⚠️ Potential is continuous; field may be discontinuous.
• ⚠️ Potential at center of ring: ( V = \frac{1}{4\pi\varepsilon_0} \frac{Q}{R} ).
• ⚠️ No work is done in moving a charge on an equipotential surface.
• ⚠️ ( \varepsilon_0 = 8.85 \times 10^{-12} ) C²/N·m² — value to verify from NCERT.
• ⚠️ Potential due to dipole: axial ( \propto 1/r^2 ), equatorial = 0.
• ⚠️ Mnemonic: “Equipotential — E-perpendicular” — remember E is perpendicular to equipotential surfaces.