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6.16 Let us take a small element dy in the loop, at a distance y from the long straight wire.

Magnetic flux associated with element dy, d $\varnothing$ = BdA, where

dA = Area of the element dy = a dy

Magnetic flux at distance y, B = $\frac{{\mu }_{0}I}{2\pi y}$ , where

I = current in the wire

${\mu }_0$ = Permeability of free space = 4 $\pi \times {10}^{-7}$ T m $A^{-1}$

Therefore,

d $\varnothing$ = $\frac{{\mu }_{0}I}{2\pi y}$ a dy = $\frac{{\mu }_{0}Ia}{2\pi }$ $\frac{dy}{y}$

$\varnothing =\frac{{\mu }_{0}Ia}{2\pi }\int \frac{dy}{y}$

Now from the figure, the range of y is x to x+a. Hence,

$\varnothing =\frac{{\mu }_{0}Ia}{2\pi }{\int }_x^{x+a}\frac{dy}{y}$ = $\frac{{\mu }_{0}Ia}{2\pi }{\left[{\log }_e?y\right]}_x^{a+x}$ = $\frac{{\mu }_{0}Ia}{2\pi }{\log }_e?(\frac{a+x}{x})$

For mutual inductance M, the flux is given as

$\varnothing =MI$ . Hence

$MI=\frac{{\mu }_{0}Ia}{2\pi }{\log }_e?(\frac{a+x}{x})$

M = $\frac{{\mu }_{0}a}{2\pi }{\log }_e?(\frac{a}{x}+1)$

Emf induced in the loop, e = Bav

= ( $\frac{{\mu }_{0}I}{2\pi x}$ ) $\times$ av

For I = 50 A, x = 0.2 m, a = 0.1 m, v = 10 m/s

e = $\frac{4\pi \times {10}^{-7}\times 50\times 0.1\times 10}{2\pi \times 0.2}$ = 5 $\times {10}^{-5}$ V

6.15 Length of the solenoid, l = 30 cm = 0.3 m

Area of cross-section, A = 25 ${cm}^2$ = 25 $\times {10}^{-4}$ $m^2$

Number of turns on the solenoid, N = 500

Current in the solenoid, I = 2.5 A

Current flowing time, t = ${10}^{-3}$ s

We know average back emf, e = $\frac{d\varnothing }{dt}$ …………………. (1)

Where $d\varnothing =$ Change in flux = NAB ……………… .(2)

B = Magnetic field strength = ${\mu }_0\frac{NI}{l}$ ………………. (3)

Where ${\mu }_0$ = Permeability of free space = 4 $\pi \times {10}^{-7}$ T m $A^{-1}$

From equation (1) and (2), we get

e = $\frac{NAB}{dt}$ = $\frac{NA}{dt}\times {\mu }_0\frac{NI}{l}$ = $\frac{{\mu }_0{N}^{2}AI}{lt}$ = $\frac{4\pi \times {10}^{-7}\times {500}^2\times 25\mathrm{}\times {10}^{-4}\times 2.5}{0.3\times {10}^{-3}}$ = 6.544 V

Hence the average back emf induced in the solenoid is 6.5 V

6.14 Length of the rod, l = 15 cm = 0.15 m

Magnetic field strength, B = 0.5 T

Resistance of the closed loop, R = 9 mΩ = 9 $\times {10}^{-3}$ Ω

Induced emf = 9 mV

Polarity of the induced emf is such that end P shows positive while end Q shows negative ends.

Speed of the rod, v = 12 cm/s = 0.12 m/s

Induced emf is given as, e = Bvl = 0.5 $\times 0.12\times 0.15$ = 9 mV

Yes, when key K is closed, excess charge is maintained by the continuous flow of current. When the key K is open, there is excess charge built up at both ends of the rods.

Magnetic force is cancelled by the electric force set-up due to the excess charge of opposite nature at both ends of the rod. There is no net force on the electrons in rod PQ when key K is open and the rod is moving uniformly. This is because magnetic force is cancelled by the electric force set-up due to the excess charge of opposite nature at both ends of the rods.

Retarding force exerted on the rod, F = IBl , where

I = Current flowing through the rod = $\frac{e}{R}$ = $\frac{9\times {10}^{-3}}{9\mathrm{}\times {10}^{-3}}$ = 1 A

F = 1 $\times 0.5\times 0.15$ = 0.075 N

Speed of the rod, v = 12 cm/s = 0.12 m/s

Power is given by P = Fv = 0.075 $\times 0.12=9\times {10}^{-3}$ W = 9 mW

When key K is open, no power is expended.

Power dissipated by heat = $I^{2}R=1\times$ 9 $\times {10}^{-3}$ W = 9 mW

The source of this power is an external agent.

In this case, no emf is induced in the coil because the motion of the rod does not cut across the field lines.

6.13 Area of the coil, A = 2 ${cm}^2$ = 2 $\times {10}^{-4}$ $m^2$

Number of turns, n = 25

Total charge flowing in the coil, Q = 7.5 mC = 7.5 $\times {10}^{-3}$ C

Total resistance of the coil and galvanometer, R = 0.50 Ω

We know, Induced current in the coil, I = $\frac{Inducedemf\left(e\right)}{R}$ ……… (1)

Induced emf is given by the equation, e = $-$ n $\frac{d\varnothing }{dt}$ ………………… (2),

Where d $\varnothing$ is the change of flux

Combining equations (1) and (2), we get

I = $\frac{-\mathrm{n}\frac{d\varnothing }{dt}}{R}$ or Idt = $-\frac{n}{R}$ d $\varnothing$ ……… (3)

Initial flux through coil, ${\varnothing }_i$ = BA, where B = Magnetic field strength

Final flux through coil, ${\varnothing }_f$ = 0

Integrating equation (3), we get

$\int Idt$ = $-\frac{n}{R}{\int }_{{\varnothing }_i}^{{\varnothing }_f}d\varnothing$

$\int Idt=Q=\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{}\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{}\mathrm{f}\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{}\mathrm{i}\mathrm{n}\mathrm{}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{}\mathrm{c}\mathrm{o}\mathrm{i}\mathrm{l}$

Therefore

Q = $-\frac{n}{R}\left({\varnothing }_f-{\varnothing }_i\right)=-\frac{n}{R}\left(0-{\varnothing }_i\right)=\frac{n{\varnothing }_i}{R}=\frac{nBA}{R}$

B = $\frac{QR}{nA}$ = $\frac{7.5\mathrm{}\times {10}^{-3}\times 0.5}{25\times 2\times {10}^{-4}}$ = 0.75 T

Hence, the field strength is 0.75 T

6.12 Side of the square loop, s = 12 cm = 0.12 m

Area of the square loop, A = 0.12 $\times 0.12=0.0144m^2$

Velocity of the loop, v = 8 cm /s = 0.08 m/s

Gradient of the magnetic field along negative x – direction.

$\frac{dB}{dx}$ = ${10}^{-3}$ T/cm = ${10}^{-1}$ T/m

Rate of decrease of magnetic field

$\frac{dB}{dt}$ = ${10}^{-3}$ T/s

Resistance of the loop, R = 4.50 mΩ = 4.5 $\times {10}^{-3}$ Ω

Rate of change of the magnetic flux due to motion of the loop in a non-uniform magnetic field is given as:

$\frac{d\varnothing }{dt}$ = A $\times \frac{dB}{dx}$ $\times v$ = $0.0144\times {10}^{-1}\times 0.08$ = 1.152 $\times {10}^{-4}$ T $m^2{s}^{-1}$

Rate of change of flux due to explicit time variation in field B is given as:

$\frac{d\varnothing \text{'}}{dt}$ = A $\times \frac{dB}{dt}$ = $0.0144\times {10}^{-3}$ = 1.44 $\times {10}^{-5}$ T $m^2{s}^{-1}$