Question

For an electric dipole in a non-uniform electric field with dipole moment parallel to the direction of the field, the force F and torque τ on the dipole respectively are______.
Fill in the blank with the correct answer from the options given below.

• A. F = 0, τ = 0
• B. F ≠ 0, τ = 0
• C. F = 0, τ ≠ 0
• D. F ≠ 0, τ ≠ 0

Answer 1

The Correct Option is B

An electric dipole in an electric field experiences forces and torques due to the interaction with the field. Let's analyze the scenarios:

Force (F) on the Dipole: The force on a dipole in a non-uniform electric field is generally not zero. This is because the strength of the electric field varies at different points in space, leading to a net force on the dipole.

Torque (τ) on the Dipole: Torque depends on the angle between the dipole moment (p) and the electric field (E). The torque τ is given by:

\(τ = pE sin θ\)

where θ is the angle between the dipole moment and the electric field. If the dipole moment is parallel to the electric field, θ is 0, making sin θ = 0. This results in:

\(τ = 0\)

Conclusion: For a dipole in a non-uniform electric field, with the dipole moment parallel to the field, the force F is non-zero (F ≠ 0), while the torque τ is zero (τ = 0).

Correct Answer: \(F ≠ 0, τ = 0\)

Answer 2

Step 1: What is an Electric Dipole?

An electric dipole consists of two equal and opposite charges, +q and -q, separated by a small distance 2l. The dipole moment p is defined as p = q * (2l) hat-p, where hat-p points from the negative to the positive charge. Here, the dipole moment is parallel to the non-uniform electric field.

Step 2: Calculate the Torque on the Dipole

The torque tau on a dipole in an electric field E is given by the cross product:

τ = p x E

The magnitude of the torque is:

τ = p * E * sin(theta)

Here, theta is the angle between p and E. Since the dipole moment is parallel to the field, theta = 0 degrees. So:

sin(0 degrees) = 0

τ = p * E * sin(0 degrees) = 0

Therefore, the torque τ = 0. This makes sense—when the dipole is aligned with the field, there’s no rotational effect!

Step 3: Calculate the Force on the Dipole

The net force on a dipole in an electric field is given by:

F = (p dot del) E

In a uniform field, E is constant, so del E = 0, and the force F = 0. But in a non-uniform field, E varies with position, so del E is not zero, leading to a net force.

Let’s use an example: Suppose the field is E = k * x i-hat, and the dipole is aligned with the field, so p = p i-hat. The gradient of the field is:

del E = partial derivative of (k * x) with respect to x i-hat = k i-hat

The force is:

F = (p dot del) E

p dot del = p partial derivative with respect to x

F = p * k i-hat

Since k is not zero, the force F is not zero. This shows that in a non-uniform field, the dipole experiences a net force (F is not zero).

Step 4: Clear Up Common Misconceptions

A common mistake is thinking the net force on a dipole is always zero. That’s only true in a uniform field! In a non-uniform field, the field strength differs at the +q and -q charges, so the forces don’t cancel out, resulting in a net force.

Step 5: Match with the Options

From our calculations:

- Torque τ = 0, because the dipole is parallel to the field.

- Force F is not zero, because the field is non-uniform.

The correct option is:

B. F is not zero, τ = 0

Final Answer

The net force on the dipole is non-zero (F is not zero) due to the non-uniform electric field, and the torque is zero (τ = 0) because the dipole moment is parallel to the field. The correct answer is B. F is not zero, τ = 0