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 $8\sqrt{3}\mathrm{Nm}$. If the dipole length is $2\mathrm{cm}$ then the charge on the dipole is $\dots \dots \dots$ c. (A) $-8\times {10}^3$ (B) $8.54\times {10}^{-4}$ (C) $8\times {10}^{-3}$ (D) $0.85\times {10}^{-6}$

Short answer

The charge on the dipole is $8\times {10}^{-3}$ C.

Step-by-step solution

Identify the given values

We are given the following values: 1. Angle between the electric dipole and the electric field θ = ${60}^{\circ }$ 2. Electric field intensity E = ${10}^5{\mathrm{NC}}^{-1}$ 3. Torque experienced by the dipole τ = $8\sqrt{3}\mathrm{Nm}$ 4. Length of the dipole d= $2\mathrm{cm}=0.02\mathrm{m}$ We need to find the charge, q, on the dipole.

Rewrite the formula for the torque

The torque experienced by a dipole in an electric field can be written as: τ = pEsinθ The dipole moment p is given by the product of the charge (q) and the distance (d) between the charges, so p = qd. Substitute this expression for the dipole moment into the torque formula: τ = qdEsinθ

Solve for the charge q

We need to solve for q in the equation: τ = qdEsinθ Divide both sides by dEsinθ to isolate q: q = $\frac{\tau }{dE\sin \theta }$ Next, substitute the given values into the equation: q = $\frac{8\sqrt{3}\mathrm{Nm}}{(0.02\mathrm{m})({10}^5{\mathrm{NC}}^{-1})\sin {60}^{\circ }}$

Calculate the charge q

We can now calculate the value of the charge on the dipole: q = $\frac{8\sqrt{3}\mathrm{Nm}}{(0.02\mathrm{m})({10}^5{\mathrm{NC}}^{-1})\left(\frac{\sqrt{3}}{2}\right)}=\frac{8\sqrt{3}\mathrm{Nm}}{0.01\mathrm{m}\cdot {10}^5{\mathrm{NC}}^{-1}\cdot \sqrt{3}}$ On simplification: q = $8\times {10}^{-3}\mathrm{C}$ Comparing this value to the given choices, we find that the correct option is (C) $8\times {10}^{-3}$ C.

Key concepts

Torque on Dipole

Torque is a measure of the force that causes an object to rotate. When an electric dipole is placed in an electric field, it can experience a torque due to the forces acting on its charges. For a dipole in an electric field, the torque $\tau$ is given by:$$\tau =pE\sin \theta$$where:

• $p$ is the dipole moment
• $E$ is the electric field intensity
• $\theta$ is the angle between the dipole moment and the electric field

This formula shows that the torque reaches its maximum value when the dipole moment is perpendicular to the electric field $(\theta ={90}^{\circ })$. If the dipole aligns with the field $(\theta =0^{\circ })$, then the torque is zero. In our exercise, the dipole is at ${60}^{\circ }$ to the field, which is a common scenario for calculating torque on a dipole in this context.

Electric Field Intensity

Electric field intensity $E$ is a measure of the strength of an electric field at a particular point. It indicates the force that a charge would experience per unit charge. Mathematically, it is expressed as:$$E=\frac{F}{q}$$where:

• $F$ is the force experienced by a small test charge
• $q$ is the magnitude of the test charge

In this exercise, the electric field intensity is given as ${10}^5\mathrm{N}/\mathrm{C}$. This high value indicates a strong electric field where significant forces can act on charges or a dipole placed within it. Understanding electric field intensity is crucial for solving problems involving forces in the field, such as the calculation of torque on a dipole.

Dipole Moment

The dipole moment $p$ is a vector quantity that represents the strength and orientation of an electric dipole. It is an important factor when analyzing torques and electric forces on a dipole in an electric field.The dipole moment is calculated by:$$p=qd$$where:

• $q$ is the magnitude of one of the charges in the dipole
• $d$ is the distance between the charges

The dipole moment's direction is from the negative charge to the positive charge. In our exercise, this relationship is critical as it links the charge on the dipole to the torque experienced.Accurate calculation of the dipole moment helps in predicting how strongly a dipole will interact with an electric field.

Charge Calculation

The process of calculating the charge $q$ on a dipole involves understanding how the torque interacts with other relevant variables.From the torque formula $\tau =qdE\sin \theta$, we rearrange to solve for the charge:$$q=\frac{\tau }{dE\sin \theta }$$By substituting the known values:

• $\tau =8\sqrt{3}\mathrm{Nm}$
• $d=0.02\mathrm{m}$
• $E={10}^5\mathrm{N}/\mathrm{C}$
• $\theta ={60}^{\circ }$

We achieve:$$q=\frac{8\sqrt{3}\mathrm{Nm}}{0.02\mathrm{m}\times {10}^5\mathrm{N}/\mathrm{C}\times \left(\frac{\sqrt{3}}{2}\right)}$$Upon simplification, we find:$$q=8\times {10}^{-3}\mathrm{C}$$This calculated charge determines how the dipole's electromagnetic properties align with the forces in an electric field.