Torque on a Dipole in a Uniform Electric Field

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Torque on a Dipole in a Uniform Electric Field

1.0What is an Electric Dipole?

An electric dipole is a pair of equal and opposite charges, +q and −q, separated by a small fixed distance, 2a. The electric dipole moment (p) is a vector quantity that measures the strength and orientation of the dipole. It is directed from the negative charge to the positive charge.

The magnitude of the electric dipole moment is given by:

p = q(2a)

2.0Torque on a Dipole in a Uniform Electric Field

When an electric dipole is placed in a uniform electric field (E), the two charges experience equal and opposite forces. The force on the positive charge, +q, is F₊=+ qE and it acts in the direction of the electric field. The force on the negative charge, −q, is F_-=- qE, and it acts opposite to the direction of the electric field.

Since these two forces are equal in magnitude (F = qE), opposite in direction, and their lines of action do not coincide, they form a couple. A couple is a pair of forces that produces a turning effect, or torque. The net force on the dipole is zero, so it will not undergo translational motion, but the net torque is non-zero, causing it to rotate.

3.0Torque on a Dipole in a Uniform Electric Field Derivation

Let's derive the formula for the torque (τ) on the dipole.

Consider an electric dipole with charges +q and −q separated by a distance 2a, placed in a uniform electric field E. The angle between the dipole moment vector p and the electric field vector E is θ.

The forces on the charges are:

Force on +q: F₊=+ qE

Force on -q: F_-=- qE

The magnitude of each force is F=qE.

Torque is defined as the force multiplied by the perpendicular distance from the pivot point to the line of action of the force. We can calculate the torque about the center of the dipole.

Torque due to force on +q; $τ_{+} = F \times a sin θ = (qE) (a sin θ)$

Torque due to force on -q: $τ_{-} = F \times a sin θ = (qE) (a sin θ)$

Both torques cause the dipole to rotate in the same direction (clockwise in the diagram above), so the total torque is the sum of the individual torques.

$Total torque: τ = τ_{+} + τ_{-} τ = (qE) (a sin θ) + (qE) (a sin θ) τ = qE (2 a) sin θ$

Since the magnitude of the dipole moment is p=q(2a), we can substitute this into the equation:

τ=pEsinθ

In vector form, the torque is the cross product of the electric dipole moment and the electric field:

τ=p×E

The direction of the torque is given by the right-hand rule and is perpendicular to both p and E. The torque tends to align the dipole moment vector with the electric field vector.

4.0Torque on a Dipole in a Uniform Electric Field Formula

Force & Torque on Dipole in a Uniform Electric Field

In uniform electric field both charges experience equal and opposite force, so net force on dipole is zero.

Torque/Couple of force = magnitude of either force × perpendicular distance between lines of force.

$τ = (qE) 1 sin θ or τ = (ql) E sin θ$

$τ = pE sin θ or τ = (p \times E)$

When dipole is aligned along the field

Here angle between EF and dipole moment is zero.

So, τ=pEsinθ = 0 (torque is zero)

When dipole is placed perpendicular to the field.

Here angle between $E & p = 9 0^{\circ} .$

So, τ=pEsinθ = pE (max) (Torque on dipole is maximum)

Work done in rotating a dipole in uniform electric field

Work done in rotating a dipole by small angle θ is

$d W = τ d θ$

Work done in rotating the dipole from initial angular position θ₁ to θ₂ is:

$\int d W = \int_{θ_{1}}^{θ_{2}} τ d θ = pE \int_{θ_{2}}^{a_{2}} sin θ d θ [∵ τ = PE sin θ]$

$W = pE [- cos θ]_{θ_{2}}^{θ_{2}} = pE (- cos θ_{2} + cos θ_{1}) = pE (cos θ_{1}, - cos θ_{2})$

Interaction Energy of a Dipole in Electric Field:

Figure shows a dipole placed at an angle θ with the direction of electric field. In this situation we consider that the two charges of dipoles are located on two equipotential surfaces having potentials V₁ and V₂ which can be related as

V₁ − V₂ = E d cosθ

The interaction energy of the dipole with electric field can be given as:

U = −qV₁ + qV₂

⇒ U = q(V₂ − V₁)

⇒ U = −qEd cosθ

⇒ U = −pE cosθ

In vector notations, we can write interaction energy of dipole in electric field from above equation as

U = −pE cosθ

(i) At θ = 0, there is minimum of P.E. so it is a stable equilibrium position.
(ii) At θ = 180°, there is maxima of P.E. so it is a position of unstable equilibrium.

The key formula you need to remember is:

Scalar form: τ = pE sinθ

Vector form: τ = p × E

Special Cases:

Maximum Torque: The torque is maximum when sinθ=1, which occurs when (dipole is perpendicular to the field).
Zero Torque: The torque is zero when sinθ=0, which occurs when θ=0⁰ or θ=180⁰ (dipole is aligned with the field).
τ=0

Potential Energy of a Dipole:

The potential energy (U) of a dipole in a uniform electric field is given by:

U = −p.E = −pE cosθ

Minimum Potential Energy: Occurs when $θ = 0^{\circ}$ (stable equilibrium), U_min = −pE
Maximum Potential Energy: Occurs when $θ = 18 0^{\circ}$ (unstable equilibrium), U_max = +pE

5.0Solved Problems

Problem 1: An electric dipole consists of two charges, ±5 nC, separated by 2 mm. It is placed in a uniform electric field of $2 \times 1 0^{5} N / C$ at an angle of $3 0^{\circ}$ . Calculate the magnitude of the torque on the dipole.

Solution:

$q = 5 \times 1 0^{- 9} C 2 a = 2 \times 1 0^{- 3} m E = 2 \times 1 0^{5} N / C θ = 3 0^{\circ}$

First, calculate the dipole moment, p:

$p = q (2 a) = (5 \times 1 0^{- 9}) (2 \times 1 0^{- 3}) = 1 \times 1 0^{- 11} C m$

Now, use the torque formula:

$τ = pE sin θ τ = (1 \times 1 0^{- 11}) (2 \times 1 0^{5}) sin (3 0^{\circ}) τ = (2 \times 1 0^{- 6}) (0.5) τ = 1 \times 1 0^{- 6} N m$

The magnitude of the torque is $1 \times 1 0^{- 6} N m$ .

Problem 2: An electric dipole is rotated from its position of stable equilibrium to unstable equilibrium. How much work is done by the external agent?

Solution:

Work done by an external agent to rotate the dipole is equal to the change in its potential energy.

$W_{e x t} = Δ U = U_{f} - U_{i}$

Initial state (stable equilibrium): $θ_{i} = 0^{\circ}, U_{i} = - pE cos (0^{\circ}) = - pE$
Final state (unstable equilibrium): $θ_{f} = 18 0^{\circ}, U_{f} = - pE cos (18 0^{\circ}) = + pE$

$W_{e x t} = (+ pE) - (- pE) = 2 pE$

The work done is 2pE.

Problem 3: A dipole is formed by two point charge -q and +q, each of mass m, and both the point charges are connected by a rod of length 'l' and mass m. This dipole is placed in uniform electric field $E$ . If the dipole is disturbed by a small angle $θ$ from stable equilibrium position, prove that its motion will be almost SHM. Also find its time period.

Solution:

If the dipole is disturbed by $θ$ angle,

$τ_{n e t} = - PE sin θ$

(Here -ve sign indicates that direction of torque is opposite to $θ$ )

If $θ$ is very small, $sin θ \approx θ$

$∴ τ_{n e t} = - (PE) θ$

$τ_{net} \propto (- θ)$ so motion will be almost SHM & C= P.E (where, P=ql)

$∵ T = 2 π \frac{I}{C}$

$∴ T = 2 π \frac{\frac{m ℓ}{12} + 2 m ( \frac{ℓ}{2} ) ^{2}}{P \cdot E} = 2 π \frac{\frac{m ℓ}{12} + \frac{m ℓ ^{2}}{2}}{q ℓ E}$

$T = 2 π \frac{7 m ℓ ^{2}}{12 q ℓ E} = 2 π \frac{7 m ℓ}{12 qE}$

$T = π \frac{7 m ℓ}{3 qE}$

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Torque on a Dipole in a Uniform Electric Field

1.0What is an Electric Dipole?

An electric dipole is a pair of equal and opposite charges, +q and −q, separated by a small fixed distance, 2a. The electric dipole moment (p) is a vector quantity that measures the strength and orientation of the dipole. It is directed from the negative charge to the positive charge.

The magnitude of the electric dipole moment is given by:

p = q(2a)

2.0Torque on a Dipole in a Uniform Electric Field

When an electric dipole is placed in a uniform electric field (E), the two charges experience equal and opposite forces. The force on the positive charge, +q, is F₊=+ qE and it acts in the direction of the electric field. The force on the negative charge, −q, is F_-=- qE, and it acts opposite to the direction of the electric field.

Since these two forces are equal in magnitude (F = qE), opposite in direction, and their lines of action do not coincide, they form a couple. A couple is a pair of forces that produces a turning effect, or torque. The net force on the dipole is zero, so it will not undergo translational motion, but the net torque is non-zero, causing it to rotate.

3.0Torque on a Dipole in a Uniform Electric Field Derivation

Let's derive the formula for the torque (τ) on the dipole.

Consider an electric dipole with charges +q and −q separated by a distance 2a, placed in a uniform electric field E. The angle between the dipole moment vector p and the electric field vector E is θ.

The forces on the charges are:

Force on +q: F₊=+ qE

Force on -q: F_-=- qE

The magnitude of each force is F=qE.

Torque is defined as the force multiplied by the perpendicular distance from the pivot point to the line of action of the force. We can calculate the torque about the center of the dipole.

Torque due to force on +q; $τ_{+} = F \times a sin θ = (qE) (a sin θ)$

Torque due to force on -q: $τ_{-} = F \times a sin θ = (qE) (a sin θ)$

Both torques cause the dipole to rotate in the same direction (clockwise in the diagram above), so the total torque is the sum of the individual torques.

$Total torque: τ = τ_{+} + τ_{-} τ = (qE) (a sin θ) + (qE) (a sin θ) τ = qE (2 a) sin θ$

Since the magnitude of the dipole moment is p=q(2a), we can substitute this into the equation:

τ=pEsinθ

In vector form, the torque is the cross product of the electric dipole moment and the electric field:

τ=p×E

The direction of the torque is given by the right-hand rule and is perpendicular to both p and E. The torque tends to align the dipole moment vector with the electric field vector.

4.0Torque on a Dipole in a Uniform Electric Field Formula

Force & Torque on Dipole in a Uniform Electric Field

In uniform electric field both charges experience equal and opposite force, so net force on dipole is zero.

Torque/Couple of force = magnitude of either force × perpendicular distance between lines of force.

$τ = (qE) 1 sin θ or τ = (ql) E sin θ$

$τ = pE sin θ or τ = (p \times E)$

When dipole is aligned along the field

Here angle between EF and dipole moment is zero.

So, τ=pEsinθ = 0 (torque is zero)

When dipole is placed perpendicular to the field.

Here angle between $E & p = 9 0^{\circ} .$

So, τ=pEsinθ = pE (max) (Torque on dipole is maximum)

Work done in rotating a dipole in uniform electric field

Work done in rotating a dipole by small angle θ is

$d W = τ d θ$

Work done in rotating the dipole from initial angular position θ₁ to θ₂ is:

$\int d W = \int_{θ_{1}}^{θ_{2}} τ d θ = pE \int_{θ_{2}}^{a_{2}} sin θ d θ [∵ τ = PE sin θ]$

$W = pE [- cos θ]_{θ_{2}}^{θ_{2}} = pE (- cos θ_{2} + cos θ_{1}) = pE (cos θ_{1}, - cos θ_{2})$

Interaction Energy of a Dipole in Electric Field:

Figure shows a dipole placed at an angle θ with the direction of electric field. In this situation we consider that the two charges of dipoles are located on two equipotential surfaces having potentials V₁ and V₂ which can be related as

V₁ − V₂ = E d cosθ

The interaction energy of the dipole with electric field can be given as:

U = −qV₁ + qV₂

⇒ U = q(V₂ − V₁)

⇒ U = −qEd cosθ

⇒ U = −pE cosθ

In vector notations, we can write interaction energy of dipole in electric field from above equation as

U = −pE cosθ

(i) At θ = 0, there is minimum of P.E. so it is a stable equilibrium position.
(ii) At θ = 180°, there is maxima of P.E. so it is a position of unstable equilibrium.

The key formula you need to remember is:

Scalar form: τ = pE sinθ

Vector form: τ = p × E

Special Cases:

Maximum Torque: The torque is maximum when sinθ=1, which occurs when (dipole is perpendicular to the field).
Zero Torque: The torque is zero when sinθ=0, which occurs when θ=0⁰ or θ=180⁰ (dipole is aligned with the field).
τ=0

Potential Energy of a Dipole:

The potential energy (U) of a dipole in a uniform electric field is given by:

U = −p.E = −pE cosθ

Minimum Potential Energy: Occurs when $θ = 0^{\circ}$ (stable equilibrium), U_min = −pE
Maximum Potential Energy: Occurs when $θ = 18 0^{\circ}$ (unstable equilibrium), U_max = +pE

5.0Solved Problems

Problem 1: An electric dipole consists of two charges, ±5 nC, separated by 2 mm. It is placed in a uniform electric field of $2 \times 1 0^{5} N / C$ at an angle of $3 0^{\circ}$ . Calculate the magnitude of the torque on the dipole.

Solution:

$q = 5 \times 1 0^{- 9} C 2 a = 2 \times 1 0^{- 3} m E = 2 \times 1 0^{5} N / C θ = 3 0^{\circ}$

First, calculate the dipole moment, p:

$p = q (2 a) = (5 \times 1 0^{- 9}) (2 \times 1 0^{- 3}) = 1 \times 1 0^{- 11} C m$

Now, use the torque formula:

$τ = pE sin θ τ = (1 \times 1 0^{- 11}) (2 \times 1 0^{5}) sin (3 0^{\circ}) τ = (2 \times 1 0^{- 6}) (0.5) τ = 1 \times 1 0^{- 6} N m$

The magnitude of the torque is $1 \times 1 0^{- 6} N m$ .

Problem 2: An electric dipole is rotated from its position of stable equilibrium to unstable equilibrium. How much work is done by the external agent?

Solution:

Work done by an external agent to rotate the dipole is equal to the change in its potential energy.

$W_{e x t} = Δ U = U_{f} - U_{i}$

Initial state (stable equilibrium): $θ_{i} = 0^{\circ}, U_{i} = - pE cos (0^{\circ}) = - pE$
Final state (unstable equilibrium): $θ_{f} = 18 0^{\circ}, U_{f} = - pE cos (18 0^{\circ}) = + pE$

$W_{e x t} = (+ pE) - (- pE) = 2 pE$

The work done is 2pE.

Problem 3: A dipole is formed by two point charge -q and +q, each of mass m, and both the point charges are connected by a rod of length 'l' and mass m. This dipole is placed in uniform electric field $E$ . If the dipole is disturbed by a small angle $θ$ from stable equilibrium position, prove that its motion will be almost SHM. Also find its time period.

Solution:

If the dipole is disturbed by $θ$ angle,

$τ_{n e t} = - PE sin θ$

(Here -ve sign indicates that direction of torque is opposite to $θ$ )

If $θ$ is very small, $sin θ \approx θ$

$∴ τ_{n e t} = - (PE) θ$

$τ_{net} \propto (- θ)$ so motion will be almost SHM & C= P.E (where, P=ql)

$∵ T = 2 π \frac{I}{C}$

$∴ T = 2 π \frac{\frac{m ℓ}{12} + 2 m ( \frac{ℓ}{2} ) ^{2}}{P \cdot E} = 2 π \frac{\frac{m ℓ}{12} + \frac{m ℓ ^{2}}{2}}{q ℓ E}$

$T = 2 π \frac{7 m ℓ ^{2}}{12 q ℓ E} = 2 π \frac{7 m ℓ}{12 qE}$

$T = π \frac{7 m ℓ}{3 qE}$

Frequently Asked Questions

What is a uniform electric field?

What is the torque on an electric dipole in a uniform electric field?

Does a dipole experience a net force in a uniform electric field?

What happens in a non-uniform electric field?

When is the potential energy of a dipole minimum?

Join ALLEN!

Torque on a Dipole in a Uniform Electric Field

1.0What is an Electric Dipole?

2.0Torque on a Dipole in a Uniform Electric Field

3.0Torque on a Dipole in a Uniform Electric Field Derivation

4.0Torque on a Dipole in a Uniform Electric Field Formula

Work done in rotating a dipole in uniform electric field

Interaction Energy of a Dipole in Electric Field:

5.0Solved Problems

Table of Contents

Frequently Asked Questions

What is a uniform electric field?

What is the torque on an electric dipole in a uniform electric field?

Does a dipole experience a net force in a uniform electric field?

What happens in a non-uniform electric field?

When is the potential energy of a dipole minimum?

Join ALLEN!

Torque on a Dipole in a Uniform Electric Field

1.0What is an Electric Dipole?

2.0Torque on a Dipole in a Uniform Electric Field

3.0Torque on a Dipole in a Uniform Electric Field Derivation

4.0Torque on a Dipole in a Uniform Electric Field Formula

Work done in rotating a dipole in uniform electric field

Interaction Energy of a Dipole in Electric Field:

5.0Solved Problems

Table of Contents