Question

The long, straight wire in the figure has a current \(i=1.00 \mathrm{~A}\) flowing in it. A square loop with 10.0 -cm sides and a resistance of \(0.0200 \Omega\) is positioned \(10.0 \mathrm{~cm}\) away from the wire. The loop is then moved in the positive \(x\) -direction with a speed \(v=10.0 \mathrm{~cm} / \mathrm{s}\) a) Find the direction of the induced current in the loop. b) Identify the directions of the magnetic forces acting on all sides of the square loop. c) Calculate the direction and the magnitude of the net force acting on the loop at the instant it starts to move.

Short answer

Answer: The direction of the induced current in the square loop is clockwise. The forces acting on each side of the loop are as follows: the upper side in the positive x-direction, the lower side in the negative x-direction, the left side in the negative y-direction, and the right side in the positive y-direction.

Step-by-step solution

Determine the Magnetic Field Direction

First, we determine the direction of the magnetic field created by the straight wire using the right-hand rule. Point the thumb in the direction of the current (positive y-direction) and curl the fingers. At the location of the loop (to the right of the wire), the magnetic field points into the page.

Apply Lenz's Law

Next, we use Lenz's Law to find the direction of the induced current in the loop. According to Lenz's Law, the induced current will flow in such direction as to create a magnetic field that opposes the change in the original magnetic field. Since the loop moves in the positive x-direction, the magnetic field lines passing through the loop are decreasing. Thus, the induced magnetic field must point into the page as well. Using the right-hand rule for the loop, the induced current flows clockwise. a) The direction of the induced current in the loop is clockwise.

Apply the Right-Hand Rule to the Loop

Now, we will apply the right-hand rule to the loop to determine the direction of the magnetic forces acting on every side of the square loop. Align your right hand with your thumb pointing in the direction of the induced current and fingers pointing in the direction of the magnetic field (into the page): 1. For the upper side, the magnetic force acts in a positive x-direction. 2. For the lower side, the magnetic force acts in a negative x-direction. 3. For the left side, the magnetic force acts in the negative y-direction. 4. For the right side, the magnetic force acts in the positive y-direction. b) The forces acting on the upper side in the positive x-direction, the lower side in the negative x-direction, the left side in the negative y-direction, and the right side in the positive y-direction.

Calculate the Net Force

To calculate the net force, we'll use the force equation for a current-carrying wire in a magnetic field: \(F = BIL\), where \(F\) is the force, \(B\) is the magnetic field strength, \(I\) is the current, and \(L\) is the length of the wire. The induced current in the square loop can be found using Ohm's law, \(EMF = IR\) where EMF is the electromotive force, I is the current, and R is the resistance of the loop. As a result, \(I = EMF/R\). The EMF induced in the square loop is given by Faraday's law of electromagnetic induction: \(EMF = \dfrac{d\phi}{dt}\), where \(\phi\) is the magnetic flux through the loop, and \(t\) is the time. Magnetic flux is given by \(\phi = BA\), where \(A\) is the area of the loop. For the given configuration, the magnetic field \(B\) is perpendicular to the area's plane. The magnetic field produced by a long straight wire is given by the equation \(B = \dfrac{-μ_{0}i}{2\pi r}\), where \(μ_0\) is the permeability of free space, \(i\) is the current flowing in the wire, and \(r\) is the distance to the wire. At the initial time, the distance to the wire is \(10.0 \mathrm{~cm}\). Now, we can find the net force acting on the loop by adding up the forces acting on each side: 1. The upper side and lower side forces are equal and opposite, so they cancel each other out. 2. The left and right side forces both act towards the current-carrying wire. To calculate the magnetic force acting on the right side, first compute the rate of change of the magnetic field as the loop moves: \(dB/dt = \dfrac{-μ_{0}i}{2\pi (10.0 \mathrm{~cm} + 10.0 \mathrm{~cm/s} \cdot t)}\cdot\dfrac{-10.0 \mathrm{~cm/s}}{(10.0 \mathrm{~cm} + 10.0 \mathrm{~cm/s} \cdot t)}\) At \(t = 0\): \(dB/dt = \dfrac{-μ_{0}i}{2\pi (10.0 \mathrm{~cm})}\cdot\dfrac{-10.0 \mathrm{~cm/s}}{(10.0 \mathrm{~cm})}\) For the right side of the square loop, the length of the wire is \(L = 10.0 \mathrm{cm}\), so the net force can be found by multiplying \(EMF = \dfrac{d\phi}{dt}\), I, and L: \(F = \dfrac{d\phi}{dt}L = \dfrac{d(BA)}{dt}L\) The magnetic force acting on the right side is: \(F = (\dfrac{-μ_{0}i}{2\pi (10.0 \mathrm{~cm})}\cdot\dfrac{-10.0 \mathrm{~cm/s}}{(10.0 \mathrm{~cm})})(A)(L)\) Assuming the left side's magnetic force is equal to the right side, the net force acting towards the current-carrying wire can be found. The magnetic force direction is towards the current-carrying wire, and at the instant it starts moving, the magnitude of the net force is: \(F = 2(\dfrac{-μ_{0}i}{2\pi (10.0 \mathrm{~cm})}\cdot\dfrac{-10.0 \mathrm{~cm/s}}{(10.0 \mathrm{~cm})})(A)(L)\) c) The net force acting on the loop at the instant it starts to move is towards the current-carrying wire with a magnitude given by the expression above.

Key concepts

Lenz's Law

Lenz's Law is a fundamental principle that describes the direction of an induced electromagnetic field (EMF) and current in response to changes in magnetic flux. The law states that an induced EMF always gives rise to a current whose magnetic field opposes the original change in magnetic flux. This is elegantly expressed as 'The induced current is in such a direction as to produce a magnetic field that opposes the change in magnetic flux'.

In the context of our exercise, when the square loop is moved towards the current-carrying wire, Lenz's Law indicates that the induced magnetic field within the loop is in the direction that opposes the increase in magnetic flux through the loop. As the loop moves closer to the wire, the magnetic field through the loop increases, and thus the induced current will flow to create an opposing magnetic field. This opposing field translates into a clockwise current for a viewer on the same side of the loop as the wire, as demonstrated in the step-by-step solution provided.

Faraday's Law of Electromagnetic Induction

Faraday's Law of Electromagnetic Induction quantifies how an electric current is induced in a conductor when it's exposed to a changing magnetic field. This law is mathematically described as \( EMF = -\frac{d\phi}{dt} \), where EMF is the induced electromotive force, and \( \frac{d\phi}{dt} \) is the rate of change of the magnetic flux through the circuit. The negative sign in this formula, introduced by Lenz's Law, indicates that the induced EMF and resulting current are directed to oppose the change in flux.

In our example, the EMF is generated as a result of the loop moving in the magnetic field. The induced EMF is then used to find the amount of current generated, taking into account the resistance of the loop. This process illustrates Faraday's Law in action, since the loop experiences a change in magnetic environment due to its motion.

Right-Hand Rule

The right-hand rule is a mnemonic device used in physics to determine the orientation of certain vector quantities in electromagnetic situations. When dealing with a current-carrying wire, if you point your thumb in the direction of the conventional current, your curving fingers show the direction of the magnetic field circling the wire. Conversely, when considering a loop or coil, if you point your curled fingers in the direction of the current, your thumb points towards the direction of the magnetic field inside the coil.

In the solved exercise, the right-hand rule is used twice: first, to determine the magnetic field around the current-carrying wire; and second, to find the direction of the magnetic forces on each side of the square loop due to the induced current. This tool makes it simpler to visualize the complex interactions between electricity and magnetism and is crucial for correctly applying other laws like Faraday's and Lenz's.

Magnetic Forces

Magnetic forces are the forces a magnetic field exerts on moving electric charges, which include the forces exerted on a current-carrying wire within a magnetic field. These forces are at the heart of the interaction described in our exercise. The force on a segment of wire carrying current \(I\) along a direction with magnetic field \(B\) is perpendicularly directed to both the field and the current, which is succinctly expressed as \(F = I(L \times B)\), where \(L\) is the length vector of the wire and \(\times\) represents the vector cross product.

The magnetic forces act on all sides of the loop and are responsible for the net force acting on the current-carrying loop. Depending on their direction and magnitude, as determined through the application of the right-hand rule, they can either add up or cancel each other out, as seen in the step-by-step solution.