Problem 70 - Master Resource Book in JEE Main Physics

Question

An electric dipole is placed at an angle of \(60^{\circ}\) with an electric field of intensity \(10^{5} \mathrm{NC}^{-1}\). It experiences a torque equal to \(8 \sqrt{3} \mathrm{~N}-\mathrm{m} .\) Calculate the charge on the dipole, if the dipole length is \(2 \mathrm{~cm}\). (a) \(-8 \times 10^{3} \mathrm{C}\) (b) \(8.54 \times 10^{-4} \mathrm{C}\) (c) \(8 \times 10^{-3} \mathrm{C}\) (d) \(0.85 \times 10^{-6} \mathrm{C}\)

Step-by-Step Solution

Answer

The charge on the dipole is \(8 \times 10^{-3} \mathrm{C}\) (Option c).

Step 1: Write down the formula for torque experienced by an electric dipole

The torque (τ) experienced by an electric dipole in a uniform electric field is given by the formula: \( \tau = pE \sin \theta \), where \( p \) is the dipole moment, \( E \) is the electric field intensity, and \( \theta \) is the angle between \( \vec{p} \) and \( \vec{E} \).

Step 2: Set up the dipole moment relation

The dipole moment \( p \) is defined as \( p = q \cdot d \), where \( q \) is the charge on each dipole end and \( d \) is the separation distance between the charges (i.e., the dipole length). For this problem, \( d = 2 \, \text{cm} = 0.02 \text{ m} \).

Step 3: Substitute known values into the torque formula

We are given \( \tau = 8 \sqrt{3} \text{ Nm} \), \( E = 10^{5} \text{ Nm/C} \), and \( \theta = 60^{\circ} \). Substitute into the torque formula: \[ 8 \sqrt{3} = q \cdot 0.02 \cdot 10^{5} \cdot \sin 60^{\circ} \].

Step 4: Calculate \( \sin 60^{\circ} \)

The value of \( \sin 60^{\circ} \) is \( \frac{\sqrt{3}}{2} \). Substitute this value into the equation from Step 3.

Step 5: Solve for the charge \( q \)

Substitute \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \) into the equation: \[ 8 \sqrt{3} = q \cdot 0.02 \cdot 10^{5} \cdot \frac{\sqrt{3}}{2} \]. Simplify to obtain: \[ 8 \sqrt{3} = q \cdot 1000 \cdot \frac{\sqrt{3}}{2} \]. Divide both sides by \( 500 \sqrt{3} \) to get: \[ q = \frac{8 \sqrt{3}}{500 \sqrt{3}} = \frac{8}{500} = 0.016 \text{ C} = 8 \times 10^{-3} \text{ C} \].

Step 6: Compare the result with the options and select the correct answer

The calculated charge is \( 8 \times 10^{-3} \text{ C} \), which corresponds to option (c) in the given choices.

Key Concepts

Torque on Dipole

When an electric dipole is placed in an electric field, it experiences a torque that tends to rotate it. This torque is a result of the forces exerted by the electric field on the positive and negative charges in the dipole. Depending on the alignment, the torque may try to align the dipole with the electric field. The formula to calculate this torque is given by:\[\tau = pE \sin \theta\]where:

• \( \tau \) is the torque experienced by the dipole.
• \( p \) is the dipole moment, which indicates the strength of the dipole.
• \( E \) is the intensity of the electric field.
• \( \theta \) is the angle between the dipole moment vector \( \vec{p} \) and the electric field vector \( \vec{E} \).

The torque reaches its maximum value when \( \sin \theta = 1 \), meaning the dipole is perpendicular to the electric field. As \( \theta \) decreases to zero, the torque also decreases, because the dipole aligns with the field, thus reducing the twisting effect.

Electric Field Intensity

The electric field intensity \( E \) is a measure of the strength of an electric field at a particular location. It is defined as the force experienced per unit positive charge placed at that point. The electric field intensity is expressed in the formula:\[E = \frac{F}{q_0}\]where:

• \( E \) is the electric field intensity.
• \( F \) represents the force experienced by a small positive test charge.
• \( q_0 \) is the magnitude of the test charge.

Electric fields are represented as vectors and they indicate both the magnitude and direction of the field. The field intensity is uniform if it has the same value at all points in the area considered. When dealing with dipoles, knowing the intensity and direction of the electric field helps predict how the dipole will behave in the field.

Dipole Moment

A dipole moment is a quantitative measure of the dipole strength. It's a vector quantity that points from the negative charge to the positive charge in a dipole. The magnitude of the dipole moment is calculated as:\[p = q \cdot d\]where:

• \( p \) is the dipole moment.
• \( q \) is the magnitude of the charge at each end of the dipole.
• \( d \) is the distance separating the charges, also known as the dipole length.

The SI unit of dipole moment is Coulomb-meter (Cm). The dipole moment helps in calculating the torque and energy of a dipole in an electric field. A larger dipole moment means a stronger dipole, which means it can interact more significantly with electric fields.