A constant current I flows along an infinitely long wire that is lined up along

Question

A constant current I flows along an infinitely long wire that is lined up along the z-axis as shown. A square loop of conducting wire lies nearby. This square loop lies in the yz-plane and has sides of length c. This loop is moving with a constant velocity v^rightarrow = v_0y^hat, with v_0 > 0. Use Ampere's law to come up with an expression for the magnetic flux density in the y-z plane as a function of y. Show that he magnetic flux through this square loop is phi = mu_0I/2 pi c ln (1 + b/c) Determine the magnitude of the electromotive force (EMF) in the loop, when it is at the position shown in the diagram. Use b=3m; c=0.5m; I = 1.3A; v = 1.2 m/s Draw an arrow on the diagram to show the direction of the induced current in the loop.

Explanation / Answer

part a:

consider an infinitely long cylinder, concentric along z axis

its radius is y.

using ampere's law,

if magnetic field intensity is H,

then integration of H.dl =current enclosed

==>H*2*pi*y=I

==>H=I/(2*pi*y)

then magnetic flux density=B=mu*H

where mu=magnetic permeability

then magnetic flux density as a function of y is given as

B=mu*I/(2*pi*y)

part b:

consider a strip of length c parallel to z axis of length dy at a distance of y from the wire

such as b<y<b+c

then magnetic flux density=mu*I/(2*pi*y)

magnetic flux=magnetic flux density*area

=mu*I*c*dy/(2*pi*y)

total magnetic flux through the square=integration of mu*I*c*dy/(2*pi*y)

from y=b to y=b+c

=(mu*I*c/(2*pi)*ln(y)

using the limits,

total flux=(mu*I*c/(2*pi))*ln((b+c)/c)

hence proved.

part c:

magnetic flux through the loop=(mu*I*c/(2*pi))*ln((b+c)/c)

magnitude of emf induced=rate of change of flux

=(mu*I*c/(2*pi))*(c/(b+c))*(1/c)*(db/dt)

here db/dt=v

so magnitude of emf induced

=(mu*I*c/(2*pi))*(v/(b+c))

using the values given,

magnitude of induced emf=(4*pi*10^(-7)*1.3*0.5/(2*pi))*(1.2/(3+0.5))

=4.4571*10^(-8) volts

part d:

as b is increasing, flux linkage is increasing in out of the page direction

so current should be such that it will oppose the originl flux. (as per lenz's law)

hence induced current will be in clockwise direction in the square loop.

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