Problem 42 - University Physics with Modern Physics

Question

Two charges, \(+e\) and \(-e,\) are a distance of \(0.68 \mathrm{nm}\) apart in an electric field, \(E,\) that has a magnitude of \(4.4 \mathrm{kN} / \mathrm{C}\) and is directed at an angle of \(45^{\circ}\) with the dipole axis. Calculate the dipole moment and thus the torque on the dipole in the electric field.

Step-by-Step Solution

Answer

Question: Calculate the dipole moment and the torque on a dipole in an electric field where the charges are +e and -e, separated by a distance of 0.68 nm, and the electric field vector has a magnitude of 4.4 kN/C and is directed at an angle of 45 degrees with the dipole axis. Answer: The dipole moment is approximately \(1.088 \times 10^{-28} \ \mathrm{C} \cdot \mathrm{m}\), and the torque on the dipole in the electric field is approximately \(3.393 \times 10^{-25} \ \mathrm{N} \cdot \mathrm{m}\).

Step 1: Calculate the dipole moment

The formula for the dipole moment is: \[\vec{p} = q\vec{d}\] where \(\vec{p}\) is the dipole moment, \(q\) is the charge, and \(\vec{d}\) is the distance between the two charges. The charges are \(+e\) and \(-e\), so the magnitude of the charge is \(e = 1.6 \times 10^{-19} \mathrm{C}\). The distance between the charges is given as \(0.68 \ \mathrm{nm}\), which is equal to \(0.68 \times 10^{-9} \ \mathrm{m}\). Therefore, the dipole moment can be calculated as: \[\vec{p} = (1.6 \times 10^{-19} \mathrm{C})(0.68 \times 10^{-9} \mathrm{m}) = 1.088 \times 10^{-28} \ \mathrm{C} \cdot \mathrm{m}\]

Step 2: Find the electric field vector

The electric field vector \(\vec{E}\) has a magnitude of \(4.4 \ \mathrm{kN}/\mathrm{C}\), which is equal to \(4.4 \times 10^{3} \ \mathrm{N}/\mathrm{C}\). The electric field is directed at an angle of \(45^{\circ}\) with the dipole axis. We can use the angle to find the electric field vector components. In the horizontal direction, \(E_{x} = E \cos(45^{\circ}) = (4.4 \times 10^{3} \ \mathrm{N}/\mathrm{C})\cos(45^{\circ})\). In the vertical direction, \(E_{y} = E \sin(45^{\circ}) = (4.4 \times 10^{3} \ \mathrm{N}/\mathrm{C})\sin(45^{\circ})\).

Step 3: Calculate the torque on the dipole

The torque on the dipole in the electric field can be calculated using the following formula: \[\vec{\tau} = \vec{p} \times \vec{E}\] Taking the cross product, we get: \[\tau_{z} = pE \sin(\theta)\] The angle \(\theta\) between the electric field vector and the dipole moment vector is \(45^{\circ}\). Using the values of dipole moment (\(p = 1.088 \times 10^{-28} \ \mathrm{C} \cdot \mathrm{m}\)) and electric field magnitude (\(E = 4.4 \times 10^{3} \ \mathrm{N}/\mathrm{C}\)), we can find the torque: \[\tau_z = (1.088 \times 10^{-28} \ \mathrm{C} \cdot \mathrm{m})(4.4 \times 10^{3} \ \mathrm{N}/\mathrm{C}) \sin(45^{\circ}) = 3.393 \times 10^{-25} \ \mathrm{N} \cdot \mathrm{m}\] The torque on the dipole in the electric field is approximately \(3.393 \times 10^{-25} \ \mathrm{N} \cdot \mathrm{m}\).

Key Concepts

Understanding Electric Charges

Electric charges are one of the fundamental properties of matter, existing in two types: positive and negative. Like charges repel each other, while opposite charges attract. The unit of charge is the coulomb (C), and the charge of a single electron (or proton) is approximately \[\begin{equation}-1.6 \times 10^{-19} \text{C (electron)}, \ 1.6 \times 10^{-19} \text{C (proton)}. \end{equation}\] The interactions between charges are described by Coulomb's law, which states the force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. This concept is crucial when analyzing the behavior of electric dipoles in an external electric field, as seen in the exercise provided.

The Role of the Electric Field

An electric field is a region around electric charges in which other charges experience a force. The strength and direction of this field are essential as they dictate how a charge will react when placed within this field. The electric field strength is measured in newtons per coulomb (N/C). An important property of electric fields is that they are vector fields, meaning they have both magnitude and direction, which was vital in solving the given exercise. The effect of an electric field on a dipole depends on its orientation relative to the field, hence, when the field is angled with respect to the dipole axis, the field's components must be considered separately to understand the total effect on the dipole.

Torque on an Electric Dipole

A torque is a measure of the force that causes an object to rotate. When an electric dipole, which consists of two equal but opposite charges separated by some distance, is placed in an electric field, the field exerts a torque on the dipole. This torque depends on the dipole moment, which is a vector quantity defined as the product of the charge magnitude and the distance between the charges, oriented from the negative to the positive charge. Mathematically, the torque on a dipole is given by the cross product of the dipole moment (\[\begin{equation}\vec{p}\end{equation}\]) and the electric field (\[\begin{equation}\vec{E}\end{equation}\]). Therefore, the resulting torque tends to align the dipole with the electric field, which is a key concept in understanding the exercise and determining the magnitude and direction of the torque that the dipole experiences in the given scenario.