Problem 57 - University Physics with Modern Physics

Question

Point charges \(q_{1}=-4.5 \mathrm{nC}\) and \(q_{2}=+4.5 \mathrm{nC}\) are separated by 3.1 mm, forming an electric dipole. (a) Find the electric dipole moment (magnitude and direction). (b) The charges are in a uniform electric field whose direction makes an angle of \(36.9^{\circ}\) with the line connecting the charges. What is the magnitude of this field if the torque exerted on the dipole has magnitude \(7.2 \times 10^{-9} \mathrm{N} \cdot \mathrm{m} ?\)

Step-by-Step Solution

Answer

(a) Electric dipole moment is \( 1.395 \times 10^{-11} \, \mathrm{C} \cdot \mathrm{m} \), directed from \( q_1 \) to \( q_2 \). (b) Electric field magnitude is \( 8597.14 \, \mathrm{N/C} \).

Step 1: Understand the Given Data and Definitions

We have two point charges forming an electric dipole: \( q_1 = -4.5 \, \mathrm{nC} \) and \( q_2 = +4.5 \, \mathrm{nC} \), separated by a distance \( d = 3.1 \, \mathrm{mm} \). An electric dipole moment \( \mathbf{p} \) is defined as the product of one of the charges and the separation distance vector pointing from negative to positive charge.

Step 2: Calculate the Magnitude of the Electric Dipole Moment

The magnitude of the electric dipole moment is \( p = q \, \times \, d \), where \( q = 4.5 \, \mathrm{nC} = 4.5 \times 10^{-9} \, \mathrm{C} \) and \( d = 3.1 \, \mathrm{mm} = 3.1 \times 10^{-3} \, \mathrm{m} \). Thus, \( p = 4.5 \times 10^{-9} \times 3.1 \times 10^{-3} = 1.395 \times 10^{-11} \, \mathrm{C} \cdot \mathrm{m} \).

Step 3: Direction of the Electric Dipole Moment

The direction of the electric dipole moment vector \( \mathbf{p} \) is from the negative charge towards the positive charge, i.e., from \( q_1 \) to \( q_2 \).

Step 4: Understand the Torque on an Electric Dipole in a Field

The torque \( \tau \) experienced by a dipole in an electric field \( \mathbf{E} \) is given by \( \tau = pE \sin\theta \), where \( \theta = 36.9^\circ \) is the angle between \( \mathbf{p} \) and \( \mathbf{E} \), and \( \tau = 7.2 \times 10^{-9} \mathrm{N} \cdot \mathrm{m} \).

Step 5: Calculate the Magnitude of the Electric Field

Rearrange the torque equation \( \tau = pE \sin\theta \) to solve for \( E \): \( E = \frac{\tau}{p \sin \theta} \). Substitute \( \tau = 7.2 \times 10^{-9} \, \mathrm{N} \cdot \mathrm{m} \), \( p = 1.395 \times 10^{-11} \, \mathrm{C} \cdot \mathrm{m} \) and \( \sin(36.9^\circ) \approx 0.6 \) to get \( E = \frac{7.2 \times 10^{-9}}{1.395 \times 10^{-11} \times 0.6} \approx 8597.14 \, \mathrm{N/C} \).

Key Concepts

Electric Dipole Moment

An electric dipole consists of two equal and opposite charges separated by a small distance. The electric dipole moment, denoted as \( \mathbf{p} \), quantifies the strength and orientation of such a dipole system. It is a vector quantity pointing from the negative charge \( q_1 \) to the positive charge \( q_2 \).

In this particular exercise, we are given two charges \( q_1 = -4.5 \, \mathrm{nC} \) and \( q_2 = +4.5 \, \mathrm{nC} \), separated by a distance \( d = 3.1 \, \mathrm{mm} \). To calculate the dipole moment, the formula is:
\[ p = q \times d \]
where \( q \) is the magnitude of one of the charges and \( d \) is the separation distance. Here, \( q = 4.5 \times 10^{-9} \, \mathrm{C} \) and \( d = 3.1 \times 10^{-3} \, \mathrm{m} \). The calculated dipole moment is:
\[ p = 4.5 \times 10^{-9} \times 3.1 \times 10^{-3} = 1.395 \times 10^{-11} \, \mathrm{C} \cdot \mathrm{m} \]
The dipole moment has both magnitude and direction, oriented from the negative to positive charge.

Torque on a Dipole

When a dipole is placed in an external electric field, it experiences a torque. This torque tries to align the dipole moment with the electric field. The torque \( \tau \) exerted on the dipole is dependent on the magnitude of the dipole moment \( p \), the strength of the electric field \( E \), and the sine of the angle \( \theta \) between the dipole moment and the field:

\[ \tau = pE \sin(\theta) \]
In the exercise, the torque is given as \( 7.2 \times 10^{-9} \, \mathrm{N} \cdot \mathrm{m} \), and the angle \( \theta = 36.9^\circ \). This angle measures the initial misalignment between the dipole and the electric field.

Understanding the role of \( \sin(\theta) \) is crucial. It indicates that the torque is zero when the dipole is aligned completely parallel or anti-parallel with the electric field (since \( \sin(0^\circ) = \sin(180^\circ) = 0 \)). Maximum torque is felt when \( \theta = 90^\circ \). Here, the actual calculation revolves around resolving the equation to find the unknown electric field.

Uniform Electric Field

A uniform electric field is one where the field strength and direction are constant at every point in the area it occupies. This means that if we place a charge anywhere within the field, the force it experiences will always be the same magnitude and direction.

In the case of our exercise, such a field affects the dipole. We find the magnitude of this uniform electric field \( E \) using the torque experienced by the dipole:
\[ E = \frac{\tau}{p \sin(\theta)} \] Here, \( \tau = 7.2 \times 10^{-9} \, \mathrm{N} \cdot \mathrm{m} \), \( p = 1.395 \times 10^{-11} \, \mathrm{C} \cdot \mathrm{m} \), and \( \sin(36.9^\circ) \approx 0.6 \). Substituting these values gives:
\[ E \approx \frac{7.2 \times 10^{-9}}{1.395 \times 10^{-11} \times 0.6} \approx 8597.14 \mathrm{N/C} \]
With this calculation, we determine the field's strength, confirming its uniform nature. A consistent electric field like this simplifies many calculations involving electric forces and torques.