Problem 55 - College Physics

Question

An electric dipole has charges of \(\pm 4.55 \mathrm{pC}\) that are separated by \(6.00 \mathrm{~cm} .\) The dipole lies on the \(x\) -axis and its center is at the origin. Located at \(y=+4.00 \mathrm{~cm}\) is a point charge carrying a charge of \(-2.50 \mathrm{pC}\). (a) Determine the net force on the dipole and its initial center of mass acceleration (including direction) if it has a total mass of \(7.25 \mathrm{ng} .\) (b) Determine the torque on the dipole about its center of mass, including direction.

Step-by-Step Solution

Answer

Net force is upward with acceleration: \( a = \frac{2 \cdot 1.98 \times 10^{-12}}{7.25 \times 10^{-12}} \). Torque points out of the page.

Step 1: Understanding the Problem

We are given an electric dipole with charges of \(4.55 \, \text{pC}\) separated by 6.00 cm along the x-axis, at the center of which is the origin. There's a point charge of \(-2.50 \, \text{pC}\) located at y = 4.00 cm. We need to find the net force on the dipole, its acceleration, and the torque about its center of mass.

Step 2: Calculating the Force on Each Dipole Charge Due to the Point Charge

Use Coulomb's law to calculate the force on each dipole charge due to the point charge. The force \(F\) is given by: \ F = \frac{k \cdot |q_1 \cdot q_2|}{r^2} \, where \(k = 8.99 \times 10^9 \, \text{N}\cdot\text{m}^2/\text{C}^2\), \(q_1 = 4.55 \, \text{pC} = 4.55 \times 10^{-12} \, \text{C}\), and \(q_2 = -2.50 \, \text{pC} = -2.50 \times 10^{-12} \, \text{C}\). The distance \(r\) from each dipole charge to the point charge is calculated using the Pythagorean theorem: \ r = \sqrt{(3.00 \, \text{cm})^2 + (4.00 \, \text{cm})^2} = \sqrt{25} = 5.00 \, \text{cm} = 0.050 \, \text{m}\.

Step 3: Determine the Net Force on the Dipole

Calculate the components of the forces on both dipole charges and find the net force. The forces \(F_+\) on the positive charge and \(F_-\) on the negative charge will have components calculated using the angle with the y-axis: \ F_x = F \cdot \frac{3}{5} \ and \ F_y = F \cdot \frac{4}{5} \. As the forces are equal and opposite in the x-direction, their net x-component is zero. The y-components add up since both are directed towards the point charge. Let this net force be \(F_{net} = 2 \, F_y\).

Step 4: Calculating the Acceleration of the Dipole's Center of Mass

The dipole's acceleration \(a\) is calculated from Newton's second law: \ F_{net} = m \cdot a \, where \(m = 7.25 \, \text{ng} = 7.25 \times 10^{-12} \, \text{kg}\). Solve for \(a\): \ a = \frac{F_{net}}{m}\.

Step 5: Determining the Torque on the Dipole

The torque \(\tau\) about the center is given by: \ \tau = d \cdot F_{\perp} \, where \(d = 0.03 \, \text{m}\) (the half distance between charges) and \(F_{\perp} = F \cdot \sin(\theta)\). We calculated earlier that \(\sin(\theta) = \frac{4}{5}\). The torque direction is determined by the right-hand rule and points out of the x-y plane if computed for the dipole's movement in the clockwise direction.

Key Concepts

Coulomb's Law

Coulomb's Law is a fundamental principle in electromagnetism that explains the force between two point charges. The law states that the force between the charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them. The formula is given by:\[ F = \frac{k \cdot |q_1 \cdot q_2|}{r^2} \]where:

• \(F\) is the magnitude of the force between the charges,
• \(k = 8.99 \times 10^9 \, \text{N}\cdot\text{m}^2/\text{C}^2\) is Coulomb’s constant,
• \(q_1\) and \(q_2\) are the values of the charges, and
• \(r\) is the separation distance between the charges.

This equation helps us determine how the proximity and strength of charges affect the force they exert on each other. For a dipole, consisting of one positive and one negative charge, these forces will contribute to the system's behavior even if the net external force cancels out in certain directions. This is crucial for calculating forces on systems like electric dipoles.

Net Force

The concept of net force involves summing up all the forces acting on an object to understand its total effect on the object’s motion. In the case of an electric dipole placed in an electric field, each charge experiences forces due to other charges around it.For the dipole in this exercise:

• The forces along the x-axis generated by the point charge on both charges of the dipole are equal in magnitude but opposite in direction, resulting in a zero net force component in the x-direction.
• The forces along the y-axis add up because they are in the same direction, resulting in a non-zero net force component in the y-direction.

The effective net force along the y-axis, labeled as \(F_{net}\), is critical for determining how the dipole will move or accelerate in the system. Understanding the net force helps predict the dipole's behavior and reaction to external electric fields.

Torque

Torque is a measure of the tendency of a force to rotate an object about an axis. It's defined as the product of the distance from the axis of rotation to where the force is applied (moment arm) and the component of the force that is perpendicular to this line. Mathematically, the torque \(\tau\) is expressed as:\[ \tau = d \cdot F_{\perp} \]where:

• \(d\) is the distance from the axis (center of mass) to the point where force is applied,
• \(F_{\perp}\) is the force component perpendicular to the line connecting the axis to the point of force application.

For an electric dipole aligned along the x-axis with external forces applied perpendicularly, this torque can make the dipole rotate about its center. The direction of the torque can be determined by the right-hand rule, and in this problem, it points out of the x-y plane when considering clockwise direction rotation. Torque plays a crucial role in understanding rotational motion and stability in physics.

Acceleration

Acceleration is the rate at which an object's velocity changes with time. In this exercise, acceleration is derived from the net force using Newton's second law, stated as:\[ F_{net} = m \cdot a \]where:

• \(F_{net}\) is the net force acting on the object,
• \(m\) is the mass of the object, and
• \(a\) is the acceleration.

For the dipole discussed, the net force causing acceleration is aligned in the y-direction. By rearranging the formula to solve for acceleration, we get:\[ a = \frac{F_{net}}{m} \]This equation shows how greater force or less mass results in higher acceleration. The initial motion of the dipole's center of mass is determined by this acceleration, informing us about the dynamics of the system under a specific force, given its mass.