You are evaluating the performance of a large electromagnet....

You are evaluating the performance of a large electromagnet. The magnetic field of the electromagnet is zero at $t=0$ and increases as the current through the windings of the electromagnet is increased. You determine the magnetic field as a function of time by measuring the time dependence of the current induced in a small coil that you insert between the poles of the electromagnet, with the plane of the coil parallel to the pole faces as in Fig. $29.5 .$ The coil has 4 turns, a radius of $0.800 \mathrm{~cm},$ and a resistance of $0.250 \Omega .$ You measure the current $i$ in the coil as a function of time $t$. Your results are shown in Fig. $\mathrm{P} 29.64 .$ Throughout your measurements, the current induced in the coil remains in the same direction. Calculate the magnetic field at the location of the coil for (a) $t=2.00 \mathrm{~s},$ (b) $t=5.00 \mathrm{~s},$ and (c) $t=6.00 \mathrm{~s}$

Key Concepts

Multiple-Turn Coil Effect

In coils with multiple turns, each turn contributes to the total induced emf. The net induced voltage is the sum of the emfs induced in each individual turn, making it essential to account for the number of turns when relating the rate of change of magnetic flux to the observable electrical response.

Ohm's Law

Ohm's Law relates the voltage across a resistor to the current passing through it and its resistance. When an emf is induced in a coil with finite resistance, this law provides the link between the induced voltage and the measurable induced current.

Magnetic Flux

Magnetic flux represents the quantity of magnetic field passing through a given area, typically defined as the product of the magnetic field strength and the component of the area perpendicular to the field. The time rate of change of magnetic flux is central to calculating induced emf in a system.

Faraday's Law of Electromagnetic Induction

This concept states that a changing magnetic flux through a circuit induces an electromotive force (emf) in the circuit. It forms the foundational principle to relate the rate of change of magnetic field and the induced voltage in coils or loops.

Lenz's Law

Lenz's Law indicates that the direction of the induced current in a circuit is such that it opposes the change in magnetic flux that produced it. This principle helps predict the direction of the induced current and ensures the conservation of energy in electromagnetic processes.

Transcript

00:01 Part a of the problem, we need to calculate the magnetic field at the location of the coil for t is equal to two seconds.

00:08 The emf induced in the coil is given by e is equal to number of terms, rate of change of magnetic flux, d5b, divided by dt, magnitude, where a phi b, the flux is equal to area times magnetic field.

00:28 Then substituting a b, we can write emf to be a times n, d, b divided by dt.

00:38 From here, then db divided by dt, the rate of change of a tick field is equal to emf divided by a times n.

00:48 The induced emf can be written in terms of induced current and the resistance.

00:52 And we can write db will be divided by dt is equal to r.

00:58 R i divided by a n then db will be equal to so d b is equal to r i divided by a times n d t integrating integrating b will be equal to integral of a d b that is integrating from zero to two seconds r i r i r i a n d d d d d d d d d d d d that is integrating from zero to two seconds r i r i a n d so integrating with respect to i, we get, so this will be r divided by a, n integrating 0 to 2 seconds, i, dt.

01:46 From the given graph we can find the value of the integration by calculating the area under the curve.

01:54 From t is equal to 0 to t is equal to 2 seconds, this will be i, integrate, of 0 to 2 idt is equal to the area that is equal to 1 over 2 2 times 3 milli -ampeer this gives us 0 .003 0mphre seconds then we substitute this value in the equation that we wrote before and find db from 0 to 2 will be 0 .0030 times r divided by an so substituting values for rn which is 0 .250 0 .250 oms times 0 .003 00 divided by by r square is the area which is r is 0 .008 .0 square times four this gives us a db from 0 to 2 to be 0 .935 tesla from 0 to 2 seconds.

03:06 Part b of the problem.

03:08 Now we calculate the magnetic field at location of the coil for t is equal to 5 seconds.

03:15 This will be equal to so db magnetic field from 0 to 5 will be equal to magnetic field from 0 to 2 plus magnetic field from 2 to 5.

03:29 So we'll calculate 2 to 5 db that is 2 to 5 will be equal to r times using the formula that we wrote previously is the area under the curve so area under the curve divided by a times n so we substitute a value 0 .250 ohms is the resistance times the area under the curve for 2 to 5 will be 3 times central power minus 3 a times three seconds divided by pi times r 0 .008 square meters times four...

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