Electric Potential and Equipotential Surfaces

Electric Potential

Electric potential is a scalar quantity that represents the amount of work needed to move a unit positive charge from a reference point (usually infinity) to a specific point in an electric field. It is a measure of the potential energy per unit charge at a point in space.

Imagine an electric field created by a source charge. If you want to bring a small positive test charge towards this source charge, you have to do work against the repulsive force (if the source charge is positive) or the attractive force (if the source charge is negative). This work done is stored as potential energy in the test charge. Electric potential is simply this potential energy divided by the magnitude of the test charge.

Potential due to a Point Charge

For a point charge 'Q' at the origin, the electric potential 'V' at a distance 'r' from the charge is given by the formula:

V = k * (Q / r)

Where:

• V is the electric potential in Volts (V).
• k is Coulomb's constant, approximately 9 x 10⁹ Nm²/C².
• Q is the magnitude of the point charge in Coulombs (C).
• r is the distance from the point charge in meters (m).

Notice that the potential is directly proportional to the charge Q and inversely proportional to the distance r. This means a larger positive charge will create a higher potential, and points farther away will have a lower potential.

Potential due to a System of Point Charges

Since electric potential is a scalar quantity, the total potential at a point due to a collection of point charges is simply the algebraic sum of the potentials due to each individual charge.

V_total = V₁ + V₂ + V₃ + ... = Σ V_i

V_total = k * Σ (Q_i / r_i)

Here, Q_i is the charge of the i-th point charge, and r_i is the distance of the i-th point charge from the point where the potential is being calculated. It is crucial to include the sign of each charge (positive or negative) in the summation.

Potential due to an Electric Dipole

An electric dipole consists of two equal and opposite charges separated by a small distance. The potential due to a dipole is more complex and depends on the position relative to the dipole.

At a point P, at a distance r from the center of the dipole, and making an angle θ with the dipole axis, the potential is given by:

V = k * (p * cos θ) / r²

Where:

• p = Q * 2a is the dipole moment (Q is the charge, 2a is the separation).
• θ is the angle between the dipole moment vector (pointing from -Q to +Q) and the position vector from the center of the dipole to point P.

Special cases:

• On the axial line (θ = 0° or 180°): V = ± k * p / r².
• On the equatorial line (θ = 90°): V = 0.

Relationship between Electric Field and Potential

Electric field and electric potential are intimately related. The electric field is the negative gradient of the electric potential. In simpler terms, the electric field points in the direction of the steepest decrease in electric potential.

E = - dV/dr (for a field along one dimension)

In three dimensions, this is expressed as:

E = - ∇V

Where ∇ is the del operator. This means:

• E_x = - ∂V/∂x
• E_y = - ∂V/∂y
• E_z = - ∂V/∂z

Conversely, the potential difference between two points A and B can be found by integrating the electric field along a path from A to B:

V_B - V_A = - ∫_A^B E ⋅ dl

This integral represents the work done by an external agent in moving a unit positive charge from A to B against the electric field.

Equipotential Surfaces

An equipotential surface is a surface on which the electric potential is constant at every point. In other words, for any two points A and B on an equipotential surface, V_A = V_B.

Consider the work done in moving a charge between two points on an equipotential surface. Since the potential is the same, the potential difference is zero.

V_B - V_A = 0

From the relationship E ⋅ dl = -dV, if dV = 0, then E ⋅ dl = 0. This implies that the electric field vector E is always perpendicular to the displacement vector dl at every point on the equipotential surface.

Properties of Equipotential Surfaces

1. No Work Done: No work is done in moving a charge between any two points on an equipotential surface. This is because the potential difference is zero.

   Work = q * (V_B - V_A) = q * 0 = 0

2. Perpendicularity to Electric Field: The electric field lines are always perpendicular to the equipotential surfaces. This is a fundamental property derived from the relationship between E and V.
3. No Intersection: Equipotential surfaces corresponding to different potentials cannot intersect. If they did, say at point P, then P would have two different potentials simultaneously, which is impossible for a single point in space.
4. Closer Spacing Indicates Stronger Field: Where equipotential surfaces are closer together, the electric field is stronger. This is because a larger change in potential occurs over a smaller distance, implying a larger electric field magnitude (E = -dV/dr).
5. Charge Distribution: For a conductor in electrostatic equilibrium, the entire conductor is an equipotential volume. Every point on the surface and inside the conductor is at the same potential. This means the electric field inside a conductor is zero, and the electric field just outside the conductor's surface is perpendicular to the surface.

Examples of Equipotential Surfaces

The shape of equipotential surfaces depends on the configuration of the source charges:

• Point Charge: The equipotential surfaces are concentric spheres centered on the point charge. The equation of these spheres is r = constant, which directly comes from V = kQ/r.
• Electric Dipole: The equipotential surfaces are more complex. They are generally closed surfaces around the dipole, becoming more spread out at larger distances. They are not spherical.
• Line Charge or Charged Wire: The equipotential surfaces are concentric cylinders with the line charge as their axis.
• Uniform Electric Field: In a uniform electric field (like that between two parallel charged plates), the equipotential surfaces are planes perpendicular to the electric field lines.

Let's visualize this. Imagine a positive point charge at the center. The equipotential surfaces are like layers of an onion, with each layer representing a specific potential value. As you move outwards, the potential decreases. The electric field lines radiate outwards from the charge and are always perpendicular to these spherical surfaces.

Now consider two parallel plates, one positively charged and one negatively charged, creating a uniform electric field between them. The electric field lines are straight and parallel, pointing from the positive plate to the negative plate. The equipotential surfaces are planes parallel to the plates, located between them. If you move from the positive plate towards the negative plate, you are moving to regions of lower potential. The electric field is perpendicular to these planar equipotential surfaces.

Potential Energy of a System of Charges

The potential energy of a system of charges is the work done to assemble the system, i.e., to bring the charges from infinity to their respective positions.

Potential Energy of Two Point Charges

To bring two charges, q₁ and q₂, from infinity to a separation distance 'r', the work done is equal to the potential energy stored in the system.

First, bring q₁ from infinity to a point. This requires no work as there are no other charges present.

Then, bring q₂ from infinity to a distance 'r' from q₁. The work done to bring q₂ is equal to the charge q₂ multiplied by the potential created by q₁ at that point.

Potential due to q₁ at distance r is V₁ = k * q₁ / r.

Work done (W) = q₂ * V₁ = q₂ * (k * q₁ / r) = k * (q₁ * q₂) / r.

This work done is stored as the potential energy (U) of the system:

U = k * (q₁ * q₂) / r

Note that if the charges are of the same sign (both positive or both negative), U is positive, indicating repulsive forces. If the charges are of opposite signs, U is negative, indicating attractive forces.

Potential Energy of a System of N Point Charges

For a system of N charges, the total potential energy is the sum of the potential energies of all possible pairs of charges.

U = Σ_{i Uij = Σi k * (qi * qj) / rij}

Where r_ij is the distance between charge q_i and charge q_j. We sum over all unique pairs (i, j) where i is less than j to avoid double counting.

Potential Energy of a Charge in an External Field

If a charge 'q' is placed at a point where the external electric potential is 'V', the potential energy of the charge at that point is given by:

U = q * V

If the charge is moved from a point A to a point B in an external field, the change in potential energy is:

ΔU = U_B - U_A = q * V_B - q * V_A = q * (V_B - V_A)

This change in potential energy is equal to the negative of the work done by the electric field, or the work done by an external agent against the electric field.

Potential Energy of an Electric Dipole in an External Field

When an electric dipole with dipole moment 'p' is placed in a uniform external electric field 'E', it experiences a torque that tends to align it with the field. The potential energy of the dipole depends on its orientation with respect to the field.

The potential energy U is given by:

U = - p ⋅ E = - pE cos θ

Where θ is the angle between the dipole moment vector 'p' and the electric field vector 'E'.

• Minimum potential energy (U = -pE) occurs when θ = 0°, i.e., the dipole is aligned with the field (stable equilibrium).
• Maximum potential energy (U = +pE) occurs when θ = 180°, i.e., the dipole is anti-aligned with the field (unstable equilibrium).
• Zero potential energy occurs when θ = 90°, i.e., the dipole is perpendicular to the field.

The work done to rotate the dipole from an initial orientation θ₁ to a final orientation θ₂ is:

W = ΔU = U_final - U_initial = - pE cos θ₂ - (- pE cos θ₁) = pE (cos θ₁ - cos θ₂)

Summary Table: Potential vs. Electric Field

Feature	Electric Field (E)	Electric Potential (V)
Nature	Vector	Scalar
Unit	N/C or V/m	Volt (V)
Origin	Force per unit charge	Potential energy per unit charge
Relation	E = -∇V	V = -∫ E⋅dl
Direction	Points from higher potential to lower potential; direction of force on a positive charge	No direction; only magnitude
Zero Potential	Can be zero if forces cancel or at infinity for some configurations	Can be zero at infinity, at specific points due to symmetry, or on equatorial lines of dipoles

Solved Example

Question: Three charges, +q, +q, and -q, are placed at the vertices A, B, and C, respectively, of an equilateral triangle of side 'a'. Calculate the potential energy of the system.

Solution: The system consists of three charges. We need to find the potential energy of three pairs: (A, B), (B, C), and (C, A).

Let the charges be q_A = +q, q_B = +q, and q_C = -q. The distances are: r_AB = a r_BC = a r_CA = a

The potential energy of the system is the sum of the potential energies of these pairs: U = U_AB + U_BC + U_CA

U_AB = k * (q_A * q_B) / r_AB = k * (+q * +q) / a = k * q² / a

U_BC = k * (q_B * q_C) / r_BC = k * (+q * -q) / a = -k * q² / a

U_CA = k * (q_C * q_A) / r_CA = k * (-q * +q) / a = -k * q² / a

Total Potential Energy U = (k * q² / a) + (-k * q² / a) + (-k * q² / a)

U = - k * q² / a

Thus, the potential energy of the system is - k * q² / a.