Problem 30 - University Physics with Modern Physics

Question

A 0.650-m-long metal bar is pulled to the right at a steady 5.0 m/s perpendicular to a uniform, 0.750 T magnetic field. The bar rides on parallel metal rails connected through a 25.0-\(\Omega\) resistor (\(\textbf{Fig. E29.30}\)), so the apparatus makes a complete circuit. Ignore the resistance of the bar and the rails. (a) Calculate the magnitude of the emf induced in the circuit. (b) Find the direction of the current induced in the circuit by using (i) the magnetic force on the charges in the moving bar; (ii) Faraday's law; (iii) Lenz's law. (c) Calculate the current through the resistor.

Step-by-Step Solution

Answer

Induced EMF is 2.44 V; current direction is opposed to bar motion and is 0.098 A through the resistor.

Step 1: Calculate Induced EMF

Use the formula for calculating the induced electromotive force (EMF) in a conductor moving perpendicularly to a magnetic field: \( \varepsilon = B \cdot v \cdot L \), where \( B = 0.750 \) T is the magnetic field strength, \( v = 5.0 \) m/s is the velocity of the bar, and \( L = 0.650 \) m is the length of the bar. Substituting the given values, \( \varepsilon = 0.750 \times 5.0 \times 0.650 = 2.4375 \) V.

Step 2: Determine Current Direction with Magnetic Force

According to the magnetic force direction on moving charges in the bar, right-hand rule suggests that the charges experience force. Fingers point in the direction of the magnetic field, and the thumb in the direction of bar's velocity, the palm points in the force on positive charges. Thus, the force pushes positive charges towards one end of the bar, indicating current flows from one rail, through the resistor, to the other rail.

Step 3: Determine Current Direction with Faraday's Law

According to Faraday's Law, a changing magnetic environment creates a current to counteract the change. The induced current direction would be such that its magnetic field opposes the motion of the bar. Knowing the resistance and using the emf's positive terminal, current flows in the circuit to oppose motion, thus confirming the same path through the resistor as indicated by the force analysis in Step 2.

Step 4: Determine Current Direction with Lenz's Law

Lenz's Law states the induced current will flow in a direction to create a magnetic field resisting the cause of its generation. From the conductor's perspective, the induced magnetic field should oppose the motion of the bar. Therefore, the current must flow in a path to produce such a magnetic field, consistent with previous analyses.

Step 5: Calculate Current through the Resistor

Use Ohm's Law to find the current through the resistor: \( I = \frac{\varepsilon}{R} \), where \( \varepsilon = 2.4375 \) V is the induced emf, and \( R = 25.0 \; \Omega \). Substituting the values: \( I = \frac{2.4375}{25.0} = 0.0975 \) A.

Key Concepts

Induced EMF

In the given situation, we have a metal bar moving through a magnetic field. This movement generates an electromotive force, or EMF, in the circuit. This occurrence is a prime example of electromagnetic induction. The induced EMF can be calculated using the formula:

• \( \varepsilon = B \cdot v \cdot L \)

where:

• \( B = 0.750 \) T is the magnetic field strength.
• \( v = 5.0 \) m/s is the velocity of the bar.
• \( L = 0.650 \) m is the length of the bar.

Plugging in these values, we compute the EMF as:

• \( \varepsilon = 0.750 \times 5.0 \times 0.650 = 2.4375 \) V

This induced EMF is responsible for driving the current through the circuit composed of the rails and resistor.

Lenz's Law

Lenz's Law relates closely to the concept of conservation of energy. It states that the direction of the induced current will always work to oppose the change in magnetic flux that produced it.
In our example, the movement of the bar introduces a change in the magnetic environment, which generates an induced current. This induced current will create its own magnetic field; according to Lenz's Law, this will act to oppose the motion of the bar.
Using the right-hand rule: If we consider the direction of the bar's motion and the magnetic field direction, Lenz's Law indicates that the current flows in the circuit in such a way that its magnetic effect opposes the bar's movement.
This is observable through the paths described, confirming that the current moves with a force opposing that of the bar's movement.

Faraday's Law

Faraday's Law of electromagnetic induction provides the foundation for understanding how the EMF is induced in a circuit. This law states that the induced EMF in any closed loop is equal to the rate of change of magnetic flux through the loop. In mathematical terms, Faraday's Law is expressed as:

• \( \varepsilon = -\frac{d\Phi}{dt} \)

where:

• \( \varepsilon \) is the induced EMF.
• \( d\Phi \) is the change in magnetic flux.

The negative sign indicates the direction of the induced EMF as per Lenz's Law. In our specific case, the changing magnetic flux is due to the motion of the bar, which translates to the continuous cutting of magnetic lines of force.
The circuit then experiences an EMF, which aligns with Faraday's principle. This results in the calculated EMF of 2.4375 V, ensuring the current flows in a manner opposing the mechanical input causing the flux alteration.