A square loop of side 20 cm starts moving at t = 0 with a velocity of 5 cm/s towards a region of uniform magnetic field as shown in the figure. Specify the time interval(s) during... A square loop of side 20 cm starts moving at t = 0 with a velocity of 5 cm/s towards a region of uniform magnetic field as shown in the figure. Specify the time interval(s) during which induced EMF is produced in the loop.
Understand the Problem
The question describes a square loop moving into a region of uniform magnetic field and asks us to find the time interval(s) during which an electromotive force (EMF) is induced in the loop. This involves understanding Faraday's law of electromagnetic induction and how it relates to changing magnetic flux through the loop.
Answer
The induced EMF is produced during the time intervals $2 \text{ s} \le t \le 6 \text{ s}$ and $14 \text{ s} \le t \le 18 \text{ s}$.
Answer for screen readers
The induced EMF is produced during the time intervals $2 \text{ s} \le t \le 6 \text{ s}$ and $14 \text{ s} \le t \le 18 \text{ s}$.
Steps to Solve
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Calculate the time it takes for the loop to enter the magnetic field. The loop needs to travel 10 cm before it starts entering the magnetic field. We can use the formula: $time = distance / velocity$ to find the time. $$t_1 = \frac{10 \text{ cm}}{5 \text{ cm/s}} = 2 \text{ s}$$
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Calculate the time it takes for the loop to completely enter the magnetic field. The loop has a width of 20 cm, so it will take some time to fully enter the magnetic field after initially entering it. $$t_2 = \frac{20 \text{ cm}}{5 \text{ cm/s}} = 4 \text{ s}$$ Therefore, the time interval during which the loop is entering the field is from $t = 2$ s to $t = 2 + 4 = 6$ s. During this time, the magnetic flux through the loop is changing, and an EMF is induced.
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Calculate the time it takes for the loop to completely pass through the magnetic field. After the loop is entirely inside the magnetic field, the magnetic flux through the loop is constant i.e. no EMF is induced. The loop will not experience a changing magnetic flux until it starts to exit the magnetic field. The length of the magnetic field region is 40 cm, so the loop has to travel 40 cm for it to start exiting the magnetic field region. $$t_3 = \frac{40 \text{ cm}}{5 \text{ cm/s}} = 8 \text{ s}$$ Therefore, the loop starts exiting the magnetic field after $2+4+8 = 14$ s.
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Calculate the time it takes for the loop to completely exit the magnetic field. Again, it will similarly take $t = 4$ s to fully exit the field once it starts exiting so the loop fully exits the field after $14 + 4 = 18$ s. Therefore the loop is exiting the field from $t = 14$ s to $t = 14 + 4 = 18$ s. During this time, the magnetic flux through the loop is changing again, and an EMF is induced.
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Specify the time intervals during which the EMF is induced. The EMF is induced when the magnetic flux is changing. This occurs when the loop is entering the field and when the loop is exiting the field. Entering: $2 \text{ s} \le t \le 6 \text{ s}$ Exiting: $14 \text{ s} \le t \le 18 \text{ s}$
The induced EMF is produced during the time intervals $2 \text{ s} \le t \le 6 \text{ s}$ and $14 \text{ s} \le t \le 18 \text{ s}$.
More Information
The induced electromotive force (EMF) is directly proportional to the rate of change of magnetic flux through the loop: $$EMF = - \frac{d\Phi}{dt}$$. When the loop is entirely within the magnetic field, the flux is constant, so no EMF is induced.
Tips
A common mistake is to only consider the time it takes for the loop to enter the field, and forget to consider the time it takes to exit the field. Also, one might think that EMF is induced throughout the time the loop is travelling in the magnetic field region, even when fully inside of it.
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