Electrostatic Potential and Capacitance: common doubts

Question 1

What is the difference between electric potential and electric potential energy?

Accepted Answer

Potential is energy per unit charge at a point, while potential energy belongs to a particular charge placed there. The potential V is the work an external agent does per unit positive charge to bring it slowly from infinity, so it depends only on the source charges. A charge q placed at that point has potential energy U = qV. Potential is measured in volts, potential energy in joules.

Question 2

Why is the work done in moving a charge in an electrostatic field independent of path?

Accepted Answer

Because the electrostatic force is conservative. The Coulomb force depends only on position and acts along the line joining the charges, so the work done between two points depends only on the start and end points. That is what lets us assign a potential to every point, with W = q(V_P − V_R). Taken round any closed path, the work done is zero.

Question 3

Why does the potential of a dipole fall as 1/r² while a point charge's falls as 1/r?

Accepted Answer

Because the potentials of the dipole's two opposite charges nearly cancel at large distance. Each charge alone gives a 1/r potential, but from far away only a small difference survives, set by the separation, giving V = p cos θ/(4πε₀r²) for r much larger than the dipole. The potential also depends on the angle θ from p, which a single point charge's potential never does.

Question 4

Why is the potential zero on the equatorial plane of a dipole while the field is not?

Accepted Answer

Every point on the equatorial plane is equally far from +q and −q, so their potentials cancel and V = 0. The field depends on how V changes from place to place, not on its value. Stepping off the plane towards either charge changes V, so the field is not zero; on the equatorial plane it points opposite to p, parallel to the dipole's axis.

Question 5

Do I add potentials as vectors when there are several charges?

Accepted Answer

No. Potential is a scalar, so the potentials due to several charges add as ordinary numbers, each with the sign of its charge: V = (1/4πε₀) Σ qᵢ/rᵢ. A negative charge contributes a negative potential, and directions play no part. This is why potential problems are usually quicker than field problems, where each contribution must be split into components before adding.

Question 6

Why is no work done in moving a charge along an equipotential surface?

Accepted Answer

Because all points on it are at the same potential, and the work done equals the charge times the potential difference, which is zero. It follows that the field has no component along the surface, so E is always perpendicular to an equipotential. Around a point charge the equipotentials are concentric spheres; in a uniform field they are parallel planes at right angles to the field.

Question 7

How are electric field and electric potential related?

Accepted Answer

The field points the way the potential falls fastest, and its size is the rate of that fall: E = −δV/δl, with δl measured perpendicular to the equipotential. Where equipotentials crowd together the field is strong; where they are far apart it is weak. The minus sign means that moving along the field always takes you to a lower potential.

Question 8

Is the electric field zero wherever the potential is zero?

Accepted Answer

No. The field depends on how quickly the potential changes, not on its value at a point. On a dipole's equatorial plane V = 0 but the field is not zero. The reverse also happens: inside a charged conducting shell E = 0, yet the potential there is not zero; it is constant and equal to the surface value. Keep the value of V and its rate of change separate.

Question 9

Why is the potential energy of two unlike charges negative?

Accepted Answer

Because they attract, so the system gives up energy as they come together from infinity. U = q₁q₂/(4πε₀r) carries the signs of the charges, so one positive and one negative charge give U < 0. Put another way, an external agent must supply positive work to pull them apart again. Like charges give positive U, since work is needed to push them together against their repulsion.

Question 10

How do I find the potential energy of a system of three or more charges?

Accepted Answer

Add one term q₁q₂/(4πε₀r) for every distinct pair, keeping the signs. Three charges make three pairs and four charges make six. Do not count any pair twice, and never pair a charge with itself. The total equals the work needed to assemble the charges from infinity, and it is the same whatever order they are brought in.

Question 11

Where is the potential energy of a dipole in a uniform field zero, and where is it least?

Accepted Answer

It is zero when the dipole is at 90° to the field and least when it points along the field. With U = −pE cos θ, the energy at θ = 0° is −pE, the stable position, and at 180° it is +pE, an unstable balance. The work needed to turn it from θ₀ to θ₁ is pE(cos θ₀ − cos θ₁), so a half turn from the stable position costs 2pE.

Question 12

Why is the electric field inside a conductor zero in electrostatics?

Accepted Answer

Because the conductor's free electrons move until they cancel any field inside it. If a field remained in the body of the conductor, it would push the free charges and they would keep moving. They stop only when the field they set up exactly balances the applied one. In the static state, then, E = 0 everywhere inside, and the whole conductor sits at one potential.

Question 13

Why does excess charge on a conductor stay only on its surface?

Accepted Answer

Since E = 0 inside the material, a Gaussian surface drawn just within it encloses no net charge, so no charge can stay in the interior. Any extra charge therefore ends up on the surface. Just outside, the field is σ/ε₀ and perpendicular to the surface; it cannot have a component along the surface, or charges there would move.

Question 14

How does electrostatic shielding work, and does it work both ways?

Accepted Answer

An empty cavity inside a conductor has zero field whatever charges or fields lie outside, because the conductor's charges rearrange to cancel them there. That is why sitting inside a car is safer in a thunderstorm. It does not work the other way: a charge placed inside the cavity still produces a field outside the conductor, since charge is induced on the outer surface.

Question 15

What is the difference between polar and non-polar dielectrics?

Accepted Answer

Non-polar molecules have no dipole moment of their own, while polar molecules have a permanent one. In a field, a non-polar molecule is stretched slightly and gains an induced dipole along the field. Polar molecules such as water already have dipoles pointing at random; the field partly lines them up against thermal motion. Either way the material gains a dipole moment per unit volume, its polarisation P.

Question 16

Why does a dielectric reduce the electric field between capacitor plates?

Accepted Answer

The polarised dielectric develops surface charges opposite in sign to the plates they face, and these weaken the field inside. Its molecular dipoles line up with the field, leaving bound negative charge next to the positive plate and bound positive charge next to the negative plate. These charges cannot leave the material, but their field opposes the plates' field, so the net field and the voltage fall.

Question 17

Why does the capacitance of a parallel plate capacitor depend only on its geometry?

Accepted Answer

Because charge and voltage rise in proportion, so their ratio C = Q/V is fixed by size, spacing and the medium. Between the plates the field is Q/(ε₀A), so V = Qd/(ε₀A) and C = ε₀A/d. Bigger plates give more capacitance and a wider gap gives less. Neither Q nor V appears in the result, so charging a capacitor more does not change its capacitance.

Question 18

What happens to charge, voltage and energy when a dielectric slab is inserted into a charged capacitor?

Accepted Answer

It depends on whether the battery is still connected. Take a slab that fills the gap completely. If the capacitor is isolated, Q stays fixed, so C rises by a factor K while V, E and U = Q²/(2C) all fall by K. If the battery stays connected, V and E are fixed, so C, Q and U = ½CV² all rise by K, the extra charge and energy coming from the battery.

Question 19

Why is the capacitance of capacitors in series less than the smallest one?

Accepted Answer

In series every capacitor holds the same charge Q, but the total voltage is shared: V = Q/C₁ + Q/C₂ + … This gives 1/C = 1/C₁ + 1/C₂ + …, which is always smaller than the smallest single value. In parallel the voltage is common and the charges add, so C = C₁ + C₂ + … These rules are the reverse of those for resistors.

Question 20

Why is the energy stored in a capacitor ½QV and not QV?

Accepted Answer

Because the voltage across the capacitor grows from zero to V while it charges, so on average each bit of charge is carried across V/2. The first charge moves through almost no voltage and the last through nearly the full V. Adding these up gives U = ½QV = ½CV² = Q²/(2C). The energy sits in the field between the plates, with energy density ½ε₀E².

Question 21

Which formula for capacitor energy should I use in a problem?

Accepted Answer

Use the form whose quantity stays fixed. If the battery remains connected, V is constant, so U = ½CV² shows at once how the energy changes with C. If the capacitor is isolated, Q is constant, so U = Q²/(2C) is the natural choice. All three forms are equal at any instant; picking the right one stops you tracking a quantity that is quietly changing.

Electrostatic Potential and Capacitance: common doubts, answered

About the chapter

Is the electric field zero wherever the potential is zero?