Three separate arrangements of vertical equipotential lines are set up in space, the potentials labelled 10 V,

NCERT Exemplar · Q3

Q. Three separate arrangements of vertical equipotential lines are set up in space, the potentials labelled 10 V, 20 V, 30 V, 40 V and 50 V from left to right. In arrangement

(i) the five lines are equally spaced; in arrangement

(ii) the 30 V and 40 V lines are bunched close together while the others are more spread out; in arrangement

(iii) the lines have yet another non-uniform spacing. In every one of the three arrangements a charged object is moved from a point A lying on the 20 V equipotential to a point B lying on the 40 V equipotential. Which statement about the work done is correct?

(a) The work done in arrangement

(i) is the greatest.

(b) The work done in arrangement

(ii) is the least.

(ii) and (iii).

(d) The work done in arrangement

(iii) is greater than in

(ii) but equal to that in (i).

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Concept understanding — Electric Potential

Electric Potential

The Idea

When you lift a book onto a shelf you do work against gravity, and the book gains gravitational potential energy that depends on where it sits. Electric charges behave the same way in an electric field. Electric potential is the electrical analogue of "height" in a gravitational field: it tells you the potential energy a unit charge would have at a given point.

Formally, the electric potential at a point is the work done in bringing a unit positive charge from infinity to that point, slowly (without giving it kinetic energy), against the electric field.

V = \frac{W}{q_0}

Here $W$ is the work done to move a small test charge $q_0$ from infinity to the point. Because both $W$ and $q_0$ are scalars, electric potential is a scalar quantity — it has magnitude but no direction. Its SI unit is the volt:

1\ \text{V} = 1\ \text{J/C}

Potential Due to a Point Charge

For a single point charge $q$ , the potential at a distance $r$ from it is

V = \frac{1}{4\pi\varepsilon_0}\,\frac{q}{r}

Notice it falls off as $1/r$ , whereas the electric field of a point charge falls off as $1/r^2$ . The potential is taken as zero at infinity, our chosen reference. A positive charge makes the potential around it positive; a negative charge makes it negative.

Potential Difference

Usually we care about the potential difference between two points $A$ and $B$ :

V_A - V_B = \frac{W_{B\to A}}{q_0}

the work per unit charge needed to move a charge from $B$ to $A$ . This is the quantity a voltmeter reads and the "voltage" that drives current in a circuit.

Relation Between Field and Potential

Field and potential are two views of the same thing. The electric field is the negative rate of change of potential with distance:

E = -\frac{dV}{dr}

The minus sign says the field points from high potential toward low potential — a positive charge, left free, rolls "downhill" in potential. Where the potential changes steeply, the field is strong.

Superposition

Because potential is a scalar, the potential due to several charges is just the algebraic sum of their individual potentials — no vectors, no angles:

V = \frac{1}{4\pi\varepsilon_0}\left(\frac{q_1}{r_1} + \frac{q_2}{r_2} + \cdots\right)

This makes potential far easier to compute than the field, which needs vector addition. Once you have $V$ everywhere, you can get the field by differentiating.

Equipotential Surfaces

An equipotential surface is a surface on which the potential has the same value everywhere.

No work is done in moving a charge along an equipotential surface (since $\Delta V = 0$ ).
The electric field is always perpendicular to an equipotential surface.
For a point charge, equipotentials are concentric spheres; for a uniform field, they are parallel planes.

Watch out

Do not confuse potential with potential energy. Electric potential $V$ is energy per unit charge (volts); electric potential energy $U = qV$ is the actual energy (joules) a charge $q$ possesses at that point.

A Quick Example

Find the potential $3\ \text{cm}$ from a charge $q = 2\ \text{nC}$ ( $\frac{1}{4\pi\varepsilon_0} = 9\times10^{9}\ \text{N·m}^2/\text{C}^2$ ):

V = \frac{(9\times10^{9})(2\times10^{-9})}{0.03} = 600\ \text{V}

The Bottom Line

Electric potential is the work per unit charge to bring a charge from infinity to a point — a scalar measured in volts. For a point charge $V = kq/r$ , potentials add as simple numbers, the field is $E = -dV/dr$ , and equipotential surfaces are always perpendicular to the field.

Electric potential and its relationship to the electric field via E = -dV/dr is one of the foundational topics of the NCERT Class 12 Physics chapter on electrostatic potential and capacitance, tested in nearly every CBSE board paper and in JEE Main/NEET. Students searching "electric potential due to point charge formula class 12 physics" will find this derivation and the equipotential-surface rules match the NCERT treatment closely.

The electrostatic force is conservative, so the work to move a charge from A to B is $W = q(V_B - V_A)$ — fixed only by the endpoint potentials. Because A is on 20 V and B is on 40 V in every arrangement, $V_B - V_A = 20\ \text{V}$ is the same, and so is the work.

Concept

Electric potential is a state function. The work done by an external agent moving charge $q$ from A to B is $W = q(V_B - V_A)$ , independent of the path and of how the equipotentials are spaced. Spacing only reflects the local field strength $E = -\dfrac{dV}{dx}$ , not the endpoint difference.

Steps

In each arrangement $V_A = 20\ \text{V}$ and $V_B = 40\ \text{V}$ .
$V_B - V_A = 40 - 20 = 20\ \text{V}$ in all three.
Hence $W = q\times 20\ \text{V}$ is identical in (i), (ii) and (iii); the different line spacings do not affect it.

Why the others fail: (a), (b) and (d) all assume the spacing (i.e. the field strength) changes the work, but the work depends only on the potential difference between A and B, which is the same everywhere here.

✓Final answer

Option (c) — the work done is the same in all three arrangements.

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Q.Three separate arrangements of vertical equipotential lines are set up in space, the potentials labelled 10 V, 20 V, 30 V, 40 V and 50 V from left to right. In arrangement

Concept understanding — Electric Potential

Electric Potential

The Idea

Potential Due to a Point Charge

Potential Difference

Relation Between Field and Potential

Superposition

Equipotential Surfaces

A Quick Example

The Bottom Line

Concept

Steps

Method: Work Done Moving a Charge Between Two Equipotential Surfaces

Steps

Q.Three separate arrangements of vertical equipotential lines are set up in space, the potentials labelled 10 V, 20 V, 30 V, 40 V and 50 V from left to right. In arrangement

Concept

Steps

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