A rectangular coil of N turns and of length a and width b is rotated at frequency f in a uniform magnetic field, as indicated in Figure. The coil is connected to co-rotating cylinders, against which metal brushes slide to make contact. (a) Show that the emf induced in the coil is given (as a function of time t) by . This is the principle of the commercial alternating-current generator. (b) What value of Nab gives an emf with when the loop is rotated at in a uniform magnetic field of 0.500 T ?
The value of
Use the expression for magnetic flux in Faraday’s law and deriving it, prove the required equation. By using the same proved equation, calculate value of Nab.
Faraday's law of electromagnetic induction states, Whenever a conductor is placed in a varying magnetic field, an electromotive force is induced in it.
Formulae are as follow:
Where is magnetic flux, B is a magnetic field, A is an area, 𝜀 is emf.
It is given that, a coil of conducting wire is placed in a magnetic field B.
The number of turns in a coil are N. The coil is rotated at frequency f.
Let the area of each turn be A.
Angular velocity of the coil is
The coil is rotated by an angle , in time t then
Magnetic flux passing through the coil is given by,
From equation 1,
Using this in the emf induced in a coil,
From equation 1,
Since, it is a rectangular coil, area
This is an expression for the induced emf in the coil at any instant t.
When = 1, the emf is maximum.
Hence, is proved.
It is given that,
Frequency of rotation,
Magnetic field, B = 0.550 T
And now find Nab.
At maximum emf, .
Using all this in equation 3,
Hence, the value of Nab is .
Therefore, use the expression for magnetic flux in Faraday’s law and by deriving it, prove the required equation. By using the same equation, calculate the value of Nab.
Question: In Figure, two straight conducting rails form a right angle. A conducting bar in contact with the rails starts at the vertex at time t = 0 and moves with a constant velocity of 5.20m/s along them. A magnetic field with B = 0.350 T is directed out of the page. (a) Calculate the flux through the triangle formed by the rails and bar at T = 3.00S. (b) Calculate the emf around the triangle at that time. (c) If the emf is , where a and n are constants, what is the value of n?
: Inductors in parallel. Two inductors L1 and L2 are connected in parallel and separated by a large distance so that the magnetic field of one cannot affect the other. (a)Show that the equivalent inductance is given by
(Hint: Review the derivations for resistors in parallel and capacitors in parallel. Which is similar here?) (b) What is the generalization of (a) for N inductors in parallel?
The figure shows two parallel loops of wire having a common axis. The smaller loop (radius r) is above the larger loop (radius R) by a distance x>>R. Consequently, the magnetic field due to the counterclockwise current i in the larger loop is nearly uniform throughout the smaller loop. Suppose that x is increasing at the constant rate . (a)Find an expression for the magnetic flux through the area of the smaller loop as a function of x. (b)In the smaller loop, find an expression for the induced emf. (c)Find the direction of the induced current.
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