1. Suppose we have a uniform magnetic field, B, directed into the plane of the d
ID: 2013752 • Letter: 1
Question
1. Suppose we have a uniform magnetic field, B, directed into the plane of the diagram below, whose magnitude varies sinusoidally as indicated. Immersed in this magnetic field is the circuit shown, which consists of a circular loop of radius r, connected to a resistor, R, through a pair of wires that are close together, and essentially out of the region of magnetic field.
a) Derive an expression for the current, I, induced in the resistor, R, as a function of time.
b) Derive an expression for the magnitude of the induced magnetic field, B, at the center of the loop as a function of time (use the Biot-Savart Law).
c) What is the amplitude of the induced magnetic field, BMAX, at the center of the loop? (when the sinusoidal time dependent term is one)
Explanation / Answer
Given:Magnetic field has the form = B = Bo sinwt
circular loop radious = r
thus,Area of the loop = A= r2
(a)
It is known by the formual , for
induced emf in the circuit is :
emf = - e = - dB /dt
= -d (BA) / dt
= A d/dt (B)
= - A w Bocoswt
If R be the resistence connected
this, induced current = I = e / R
= (-A w cos Bo wt ) /(R )
thus,induced current : I = (A w Bo /R ) cos wt
I = r2 (w Bo /R ) cos wt
(this is in the function in the time )
(b)
Thus, Magnitude of the Magnetic field due this induced
currrent circulates in the loop given by the formual as ;
Induced Magnetic field : B = o I / 2 r
B = o I / 2 r
= ( o r2 w Bo / 2 r R ) cos wt
= ( o rw Bo / 2 R ) cos wt
(c)
Maximum induced emf :
BMAX = ( o A w Bo / 2 r R ) cos wt
If time depending term is 1
i.e coswt = 1
then , BMAX = ( o rw Bo / 2 R )
Magnetic field has the form = B = Bo sinwt circular loop radious = r thus,Area of the loop = A= r2
(a)
It is known by the formual , for
induced emf in the circuit is :
emf = - e = - dB /dt
= -d (BA) / dt
= A d/dt (B)
= - A w Bocoswt
If R be the resistence connected
this, induced current = I = e / R
= (-A w cos Bo wt ) /(R )
thus,induced current : I = (A w Bo /R ) cos wt
I = r2 (w Bo /R ) cos wt
(this is in the function in the time )
(b)
Thus, Magnitude of the Magnetic field due this induced
currrent circulates in the loop given by the formual as ;
Induced Magnetic field : B = o I / 2 r
B = o I / 2 r
= ( o r2 w Bo / 2 r R ) cos wt
= ( o rw Bo / 2 R ) cos wt
(c)
Maximum induced emf :
BMAX = ( o A w Bo / 2 r R ) cos wt
If time depending term is 1
i.e coswt = 1
then , BMAX = ( o rw Bo / 2 R )
(a) It is known by the formual , for induced emf in the circuit is : emf = - e = - dB /dt = -d (BA) / dt = A d/dt (B) = - A w Bocoswt If R be the resistence connected this, induced current = I = e / R = (-A w cos Bo wt ) /(R ) thus,induced current : I = (A w Bo /R ) cos wt I = r2 (w Bo /R ) cos wt (this is in the function in the time ) (b) Thus, Magnitude of the Magnetic field due this induced currrent circulates in the loop given by the formual as ; Induced Magnetic field : B = o I / 2 r B = o I / 2 r = ( o r2 w Bo / 2 r R ) cos wt = ( o rw Bo / 2 R ) cos wt (c) Maximum induced emf : BMAX = ( o A w Bo / 2 r R ) cos wt If time depending term is 1 i.e coswt = 1 then , BMAX = ( o rw Bo / 2 R )
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