Problem 70 - University Physics with Modern Physics

Question

A square, conducting, wire loop of side L, total mass m, and total resistance R initially lies in the horizontal xy-plane, with corners at (\(x, y, z\)) = (0, 0, 0), (0, \(L\), 0), (\(L\), 0, 0), and (\(L, L\), 0). There is a uniform, upward magnetic field \(\overrightarrow{B}\) = B\(\hat{k}\) in the space within and around the loop. The side of the loop that extends from (0, 0, 0) to (\(L\), 0, 0) is held in place on the \(x\)-axis; the rest of the loop is free to pivot around this axis. When the loop is released, it begins to rotate due to the gravitational torque. (a) Find the \(net\) torque (magnitude and direction) that acts on the loop when it has rotated through an angle \(\phi\) from its original orientation and is rotating downward at an angular speed \(\omega\). (b) Find the angular acceleration of the loop at the instant described in part (a). (c) Compared to the case with zero magnetic field, does it take the loop a longer or shorter time to rotate through 90\(^\circ\) ? Explain. (d) Is mechanical energy conserved as the loop rotates downward? Explain.

Step-by-Step Solution

Answer

(a) \( \tau_{\text{net}} = \left(\frac{B^2 L^3 \omega}{2R} - \frac{mgL}{2}\right) \sin \phi \). (b) \( \alpha = \frac{3}{2L}\left( \frac{B^2 L^3 \omega}{2R} - \frac{mgL}{2} \right) \sin \phi \). (c) Longer time due to magnetic resistance. (d) No, mechanical energy is not conserved.

Step 1: Analyze Gravitational Torque

The loop rotates due to gravitational torque. When the loop rotates by angle \( \phi \), the center of mass is at height \( \frac{L}{2} \sin \phi \). The gravitational force \( mg \) acts downward. The torque due to gravity is \( \tau_g = -mg \times \frac{L}{2} \sin \phi \). The negative sign indicates that gravity acts to rotate the loop in the clockwise direction.

Step 2: Consider Magnetic Torque

As the loop rotates, a current is induced due to the changing magnetic flux. The electromotive force (emf) induced is \( \mathcal{E} = -\frac{d\Phi}{dt} = BL^2 \omega \sin \phi \) by Faraday's Law. The current is \( I = \frac{BL^2 \omega \sin \phi}{R} \). The magnetic force interacts with the magnetic field generating a torque \( \tau_{ ext{mag}} = ILB \times \frac{L}{2} = \frac{B^2 L^3 \omega \sin \phi}{2R} \), counteracting gravity.

Step 3: Determine Net Torque

The net torque \( \tau_{ ext{net}} \) is the sum of gravitational and magnetic torques: \( \tau_{ ext{net}} = \tau_g + \tau_{ ext{mag}} = -mg \times \frac{L}{2} \sin \phi + \frac{B^2 L^3 \omega \sin \phi}{2R} \). This gives \( \tau_{\text{net}} = (\frac{B^2 L^3 \omega}{2R} - \frac{mgL}{2}) \sin \phi \). The direction of \( \tau_{\text{net}} \) depends on which term dominates.

Step 4: Calculate Angular Acceleration

Angular acceleration \( \alpha \) is related to torque by \( \tau = I\alpha \), where \( I \) is the moment of inertia of the loop about the axis of rotation. For a square loop pivoting around one edge, \( I = \frac{1}{3}mL^2 \). Thus, \( \alpha = \frac{\tau_{\text{net}}}{I} = \frac{(\frac{B^2 L^3 \omega}{2R} - \frac{mgL}{2}) \sin \phi}{\frac{1}{3}mL^2} = \frac{3}{2L}(\frac{B^2 L^3 \omega}{2R} - \frac{mgL}{2}) \sin \phi \).

Step 5: Compare Rotation Time with and Without Magnetic Field

Without a magnetic field, the loop only experiences gravitational torque, simplifying the motion. The presence of a magnetic field adds an additional force that opposes gravity, thus increasing the net torque and reducing the angular acceleration, resulting in longer time to rotate through 90°.

Step 6: Examine Mechanical Energy Conservation

The mechanical energy includes potential, kinetic, and magnetic energy. With EMF introduction and induced current, there is energy dissipation as heat (I^2R loss). Hence, mechanical energy is not conserved during rotation due to generation of heat in the resistor.

Key Concepts

Gravitational Torque

Gravitational torque is a force that causes an object to rotate around a pivot point due to gravity pulling down on the object's mass. In our scenario, the square loop of wire experiences gravitational torque as it pivots around the x-axis.Let's break it down: the loop's center of mass is positioned at a distance of \( \frac{L}{2} \sin \phi \) from the pivot point when the loop rotates through an angle \(\phi\). This causes a gravitational force \( mg \) to act downward on the loop's mass \( m \).
The gravitational torque \( \tau_g \) can be defined in mathematical terms as \(-mg \times \frac{L}{2} \sin \phi \). The negative sign represents that this torque tends to rotate the loop in a clockwise direction.
Understanding gravitational torque in this setup helps us recognize how gravity affects the motion of the loop as it tries to make it fall downward under Earth's pull.

Magnetic Torque

Magnetic torque arises when a magnetic field interacts with a current flowing through the conductor. This happens in our loop as it rotates under the influence of a magnetic field.The induced current is a result of changing magnetic flux through the loop, dictated by Faraday's Law. The electromotive force (emf) induced is given by \( \mathcal{E} = -\frac{d\Phi}{dt} = BL^2 \omega \sin \phi \). By Ohm's Law, the current \( I \) through the loop can be determined as:

• \( I = \frac{BL^2 \omega \sin \phi}{R} \)

This induced current then interacts with the magnetic field, generating a torque, iclculated as \( \tau_{\text{mag}} = ILB \times \frac{L}{2} \). Simplifying, it becomes \( \frac{B^2 L^3 \omega \sin \phi}{2R} \).
This magnetic torque acts to oppose the gravitational torque, thus it plays a role in altering how fast or slow the loop rotates in response to gravity.
In essence, magnetic torque provides an opposing force to the gravitational torque, impacting the loop's motion by affecting the overall net torque acting on the system.

Angular Acceleration

Angular acceleration tells us how quickly an object's rotational speed changes over time. For our rotating loop, it is influenced by the net torque acting on it.The net torque \( \tau_{\text{net}} \) is the sum of gravitational and magnetic torques, given by:

• \( \tau_{\text{net}} = \left( \frac{B^2 L^3 \omega}{2R} - \frac{mgL}{2} \right) \sin \phi \)

The angular acceleration \( \alpha \) is related to torque through the moment of inertia \( I \) of the loop. For a square loop pivoting about one of its edges, \( I = \frac{1}{3}mL^2 \). Thus, angular acceleration is calculated as:
\[ \alpha = \frac{\tau_{\text{net}}}{I} = \frac{3}{2L}\left( \frac{B^2 L^3 \omega}{2R} - \frac{mgL}{2} \right) \sin \phi \]
Understanding angular acceleration helps predict how quickly the loop rotates, reflecting the competitive influences of gravitational and magnetic torques on the system.

Conservation of Energy

The concept of energy conservation is vital in understanding motion and dynamics of a system. Generally, the total mechanical energy in systems without non-conservative forces remains constant over time. However, in our setup, this is not the case.While the loop rotates, the presence of an inductive emf leads to current flow through its resistance \(R\), resulting in energy dissipated as heat. This phenomenon is known as Joule heating, quantified by \( I^2R \) losses.
This energy loss indicates that mechanical energy isn't fully conserved; instead, some of it is transformed into thermal energy due to resistance in the wire.

• This energy transformation impacts the system as it changes from potential and kinetic forms into heat, causing energy to be transferred out of the mechanical system.
• It provides a crucial understanding of how energy dynamics work when additional electromagnetic forces (like those inducing current) come into play.

In essence, while gravitational and magnetic influences act on the loop, energy is not conserved due to these continuous transformations and losses.