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Chapter 27: Problem 41

A uniform magnetic field is given by \(\vec{B}=b t \hat{k},\) where \(b=\) 0.35 T/s. Find the induced current in a conducting loop with area \(240 \mathrm{cm}^{2}\) and resistance \(0.20 \Omega\) that lies in the \(x\) -y plane. In what direction is the current, as viewed from the positive z-axis?

Short Answer

Expert verified
The induced current in the loop is \( -0.42 \) A, and it is flowing in a counterclockwise direction when viewed from the positive z-axis.

Step by step solution

01

Calculate the rate of change of magnetic flux

First, calculate the magnetic flux \( \Phi \) using the equation for the magnetic flux \( \Phi = B \cdot A \), where \( B \) is the magnetic field and \( A \) is the area. The magnetic field is given as \( B = b \cdot t \), where \( b = 0.35 \) T/s and \( t \) is time. Therefore, the change in magnetic flux \( \Delta \Phi \) over time \( \Delta t \) is \( \Delta \Phi = b \cdot A \cdot \Delta t = b \cdot A \) as \( \Delta t \) equals to 1 second by default.
02

Calculate the induced emf

The induced emf \( \varepsilon \) can be calculated from the change in the magnetic flux over time, using Faraday's law \( \varepsilon = - \frac{\Delta \Phi}{\Delta t} \). Since \( \Delta t \) is one second, the induced emf is \( \varepsilon = - b \cdot A \). The negative sign indicates that the induced emf will act to oppose the change in the magnetic flux, following Lenz's law.
03

Calculate the induced current

Next, calculate the induced current \( I \) using Ohm's law \( I = \frac{\varepsilon}{R} \), where \( \varepsilon \) is the induced emf and \( R \) is the resistance of the loop. The resistance of the loop is given as \( R = 0.20 \) Ohm.
04

Determine the direction of the induced current

The direction of the induced current can be determined based on Lenz's law, which states that the induced current will flow in such a way as to oppose the change in the magnetic flux. In this case, since the magnetic field is increasing in the positive z-direction, the induced current will create its own magnetic field in the opposite direction. Hence, the current should flow in a counterclockwise direction when viewed from the positive z-axis.

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