Question

A long, straight wire has a 10.0 - A current flowing in the positive \(x\) -direction, as shown in the figure. Close to the wire is a square loop of copper wire that carries a 2.00 - A cur- rent in the direction shown. The near side of the loop is \(d=0.50 \mathrm{~m}\) away from the wire. The length of each side of the square is \(a=1.00 \mathrm{~m}\). a) Find the net force between the two current-carrying objects. b) Find the net torque on the loop.

Short answer

Answer: The net force between the two current-carrying objects is 0 N, and the net torque on the loop is 0 Nm.

Step-by-step solution

Find the magnetic field at the position of the square loop

To find the magnetic field created by the long, straight wire, we'll use Ampere's law. The magnetic field at a distance r from an infinite straight wire carrying current I can be found using the formula: \(B=\dfrac{\mu_0 I}{2\pi r}\) Given, current in straight wire (I1) = 10.0 A, and the distance (d) of the near side of the loop from the wire = 0.50 m. The magnetic field (B) can be calculated using the following values: \(B =\dfrac{\mu_0 (10.0)} {2\pi (0.5)}\)

Calculate the force on each side of the square loop

Now, we will calculate the force on each side of the square loop due to the magnetic field created by the straight wire. The magnetic force on a current-carrying wire can be calculated using the formula: \(F = I_2LB\) Where I2 is the current in the loop, L is the length of the wire in the loop, and B is the magnetic field strength at the position of the loop. Given I2 = 2.00 A and L = 1.00 m, we can calculate the force on each side of the loop. \(F = (2.00)(1.00)B\)

Find the net force on the loop

The net force between the two current-carrying objects can be found by summing up the force on each side of the square loop. Since the forces on opposite sides of the loop are equal in magnitude and opposite in direction, they will cancel each other out. Therefore, the net force on the loop will be zero. Net Force = 0 N

Find the torque on the loop

To calculate the torque on the loop, we can use the following formula: Torque = Force × Distance × sin(theta) Where distance is the distance between the straight wire and the midpoint of the side of the loop, and theta is the angle between the force and the distance. Since the torque on opposite sides of the loop will have opposite directions, they will cancel each other out. Therefore, the net torque on the loop will be zero. Net Torque = 0 Nm To summarize, the net force between the two current-carrying objects is 0 N, and the net torque on the loop is 0 Nm.

Key concepts

Ampere's Law

Ampere's Law is pivotal for understanding how, and why, a magnetic field is generated around currents. In essence, this law relates the integrated magnetic field around a closed loop to the electric current passing through the loop. The mathematical expression for Ampere's Law is:

\[ \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}} \]

where \( \vec{B} \) is the magnetic field, \( d\vec{l} \) is a differential element of the closed path, \( \mu_0 \) is the permeability of free space, and \( I_{\text{enc}} \) is the total current enclosed by the path. For simpler cases, like the one involving a long, straight wire, the magnetic field can be derived directly leading to an expression:\[ B = \frac{\mu_0 I}{2\pi r} \]

This formula gives the magnetic field at a distance 'r' away from a straight conductor carrying a current 'I'. Ampere's Law helps students understand that the magnetic field's presence is exclusively due to moving charges (currents), an insight described by directly observing the relationship between them.

Magnetic Field

The magnetic field is a fundamental concept in electromagnetism, representing the region around a magnet or a current-carrying conductor where forces due to magnetism can be felt. It is a vector field, which means it has both a direction and magnitude and is denoted by \( \vec{B} \).

When dealing with a long, straight wire carrying a steady current, as in the original exercise, the magnetic field can be visualized as concentric circles around the wire. The direction of the magnetic field lines obeys the right-hand rule: if you point the thumb of your right hand in the direction of the current, your fingers will curl in the direction of the magnetic field lines.

In practical applications and exercises, the magnitude of the magnetic field can be crucial for calculating forces on other currents or magnetic materials within that field, as depicted in problems that require determining the force on another current-carrying conductor in proximity to the wire.

Force on Current-Carrying Conductor

A key phenomenon in electromagnetism is the force exerted on a current-carrying conductor placed within a magnetic field. This interaction is the foundation for many devices, from electric motors to measuring instruments. The force is given by the equation:

\[ \vec{F} = I (\vec{L} \times \vec{B}) \]

where \( \vec{F} \) is the force vector, 'I' is the current, \( \vec{L} \) is the length vector of the conductor (with direction consistent with the current flow), and \( \vec{B} \) is the magnetic field. The cross product indicates that the force is perpendicular to both the length of the conductor and the magnetic field.

For a simple, straight conductor, the direction of this force can also be determined using a right-hand rule: point your thumb in the direction of the current ('I') and your index finger in the direction of the magnetic field ('B'). Your middle finger, at a right angle to the other two, will point in the direction of the force ('F'). This practical approach aids in understanding how, for instance, the sides of the wire loop in the exercise experience forces that can either attract or repel each other, depending on their relative directions of current flow.

Torque on Current Loop

Torque in the context of a current loop placed in a magnetic field expresses the rotational effect experienced by the loop. This is a key concept that explains the operation of many electromagnetic systems like motors and generators. The torque on a rectangular loop of wire with sides 'a' and 'b', carrying current 'I' in a magnetic field \( \vec{B} \) is given by:

\[ \tau = IaBb \sin(\theta) \]

where \( \tau \) is the torque exerted on the loop, and \( \theta \) is the angle between the plane of the loop and the magnetic field. If the loop is square, as in the exercise, 'a' and 'b' would simply be the same length, representing the sides of the square.

In situations where the forces are symmetrical, as in the square loop in proximity to the straight wire from the original problem, the torques exerted on opposite sides can cancel each other resulting in a net torque of zero. Understanding the orientation of the loop relative to the magnetic field and its implications on the net torque is essential for solving practical problems involving rotating currents in magnetic fields.