91Ó°ÊÓ

CALC A Changing Magnetic Field. You are testing a new data-acquisition system. This system allows you to record a graph of the current in a circuit as a function of time. As part of the test, you are using a circuit made up of a 4.00 -cm-radius, 500 -turn coil of copper wire connected in series to a \(600-\Omega\) resistor. Copper has resistivity \(1.72 \times 10^{-8} \Omega \cdot \mathrm{m},\) and the wire used for the coil has diameter 0.0300 \(\mathrm{mm}\) . You place the coil on a table that is tilted \(30.0^{\circ}\) from the horizontal and that lies between the poles of an electromagnet. The electromagnet generates a vertically upward magnetic field that is zero for \(t<0,\) equal to \((0.120 \mathrm{T}) \times\) \((1-\cos \pi t)\) for \(0 \leq t \leq 1.00 \mathrm{s},\) and equal to 0.240 T for \(t>1.00 \mathrm{s}\) . (a) Draw the graph that should be produced by your data-acquisition system. (This is a full-featured system, so the graph will include labels and numerical values on its axes.) (b) If you were looking vertically downward at the coil, would the current be flowing clockwise or counterclockwise?

Step by step solution

01

Calculate the Induced EMF for 0 ≤ t ≤ 1s

For the time interval \(0 \leq t \leq 1\,\mathrm{s}\), the magnetic field is given by \(B(t) = 0.120(1 - \cos(\pi t))\). The induced EMF in the coil can be calculated using Faraday's law: \(\varepsilon = -N \frac{d\Phi}{dt}\), where \(\Phi = BA\cos(\theta)\) is the magnetic flux, \(N\) is the number of turns, \(A\) is the area of the coil, and \(\theta\) is the angle between the field line and the normal to the coil, which is \(30^{\circ}\). The coil's radius is 0.04 m, so \(A = \pi (0.04)^2\). First, find \(\Phi(t)\) and then \(\varepsilon(t)\).

02

Calculate the Current in the Circuit for 0 ≤ t ≤ 1s

The current \(I(t)\) in the circuit can be given by Ohm's law: \(I = \frac{\varepsilon}{R}\), where \(R = 600\,\Omega\) is the resistance of the circuit. Using the expression for \(\varepsilon(t)\) derived in Step 1, calculate \(I(t)\) for \(0 \leq t \leq 1\,\mathrm{s}\).

03

Analyze the Magnetic Field for t > 1s

For \(t > 1\,\mathrm{s}\), the magnetic field \(B(t)\) is constant at 0.240 T. Since the magnetic field is constant, \(\frac{d\Phi}{dt} = 0\), and thus no EMF is induced. Consequently, the current is zero for \(t > 1 \mathrm{s}\).

04

Draw the Graph

Based on Steps 1 and 2, plot the current \(I(t)\) against time \(t\). For \(0 \leq t \leq 1\,\mathrm{s}\), \(I(t)\) is derived from the induced EMF. For \(t > 1\,\mathrm{s}\), \(I(t) = 0\). Label the axes with time (s) on the x-axis and current (A) on the y-axis.

05

Determine the Direction of Current Flow

Using the right-hand rule and considering the coil's setup, evaluate the direction of the induced current. Since the magnetic field increases in the positive \(\hat{y}\)-direction, the induced current will flow in a direction to oppose this change according to Lenz's Law. Looking from above, the current should be flowing counterclockwise.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Induced EMF

Faraday's Law of Induction provides the foundation for understanding how an induced EMF, or electromotive force, is generated in a circuit. This law states that an EMF is produced when there is a change in magnetic flux, and it is calculated using the equation: \[\varepsilon = -N \frac{d\Phi}{dt}\]This equation indicates that the EMF (\varepsilon) depends on the rate of change of the magnetic flux (\Phi) with time, where \(N\) is the number of turns in the coil.
The negative sign in the equation signifies Lenz's Law, which tells us that the induced EMF will always act to oppose the change in the magnetic field that produced it. For example, if the magnetic field is increasing, the induced EMF will generate a current whose magnetic field opposes this increase. This concept is critical for accurately determining the direction and magnitude of the current in the circuit during its operation.

Magnetic Flux

Magnetic flux is a measure of the amount of magnetic field passing through a given area, and it is crucial for understanding electromagnetic induction. It can be thought of as the 'flow' of the magnetic field through a surface and is calculated using the formula:\[\Phi = BA\cos(\theta)\]where \(B\) is the magnetic field strength, \(A\) is the area over which it is acting, and \(\theta\) is the angle between the magnetic field lines and the normal to the surface.
In this exercise, the coil lies at an angle to the magnetic field, and we must consider this angle to calculate the correct magnetic flux. This understanding helps one to appreciate how changing any of these parameters—like the angle, the area of the coil, or the strength of the magnetic field—affects the magnetic flux, and consequently, the induced EMF in an electrical circuit.

• Area (\(A = \pi (0.04)^2\)): Determined using the radius of the coil.
• Angle (\(\theta = 30^{\circ}\)): Influences the effective magnetic field through the coil.

Lenz's Law

Lenz's Law is pivotal in determining the direction in which induced currents flow. It states that the direction of the induced current will be such that it opposes the change in magnetic flux that produced it. This is often remembered with the right-hand rule, a handy mnemonic:

• Point the thumb of your right hand in the direction of the magnetic field increase.
• Curl your fingers—they point the direction of the induced current.

Applying Lenz's Law to the exercise, since the magnetic field is increasing upward, the induced current will create its magnetic field in the opposite direction, as seen from above the coil. Thus, when observed vertically downward, the current flows counterclockwise. Lenz's Law ensures conservation of energy by opposing changes that would increase energy unboundedly in a closed loop system.

Electrical Circuit Analysis

Electrical circuit analysis involves using principles like Ohm's Law and Kirchhoff's Laws to understand electrical systems. In this particular exercise, once the induced EMF is calculated, Ohm's Law (\(I = \frac{\varepsilon}{R}\)) becomes the tool used to find the current (\(I\)) in the circuit.

• Resistance (\(R = 600 \, \Omega\)):
The resistance in the wire and resistor affects current flow.
• Induced EMF dependency:
As EMF varies over time, so does the current.

In the interval where the magnetic field is increasing, the changing EMF generates a current through the circuit by overcoming the resistance. Beyond 1 second, when the magnetic field is constant, the EMF drops to zero; consequently, the current also becomes zero. This aspect of electrical circuit analysis highlights the dynamic behavior of real-world circuits in response to fluctuating magnetic fields, important for designing circuits and understanding transients in electrical engineering.