question_answer

(i) Explain the meaning of the term mutual inductance. Consider two concentric circular coils, one of radius \[{{r}_{1}}\] and the other of radius \[{{r}_{2}}\,\,({{r}_{1}}<{{r}_{2}})\] placed co-axially with centres coinciding with each other. Obtain the expression for the mutual inductance of the arrangement.

(ii) A rectangular coil of area A, having number of turns N is rotated at \['f'\] revolutions per second in a uniform magnetic field B, the field being perpendicular to the coil. Prove that the maximum emf induced in the coil is \[2\pi fNBA.\]

Or

Define mutual inductance of a pair of coils and write on which factors does it depends.

A square loop of 20 cm is initially kept 30 cm away from a region of uniform magnetic field of 0.1 T as shown in figure. It is then moved towards the right with a velocity of \[10\,c{{m}^{-1}}\] till it goes out of the field. Plot a graph showing the variation of

(i) Magnetic flux \[(\phi )\] through the loop with time (t)

(ii) Induced emf (e) in the loop with time (t)

(iii) Induced current in the loop if it has resistance of \[0.1\,\Omega .\]

Answer:

(i) Mutual inductance of the two coils is numerically equal to the magnetic flux with one coil when a unit current flows through the neighbouring coil. Consider two concentric circular coils \[{{C}_{1}}\] and \[{{C}_{2}},\]one of inner solenoid \[{{r}_{1}}\] and the outer solenoid \[{{r}_{2}}.\] Suppose that a current \[{{I}_{2}}\] is passed through the coil \[{{C}_{2}},\] then the magnetic field produced by this coil at its centre, \[{{B}_{2}}=\frac{{{\mu }_{0}}}{4\pi }.\frac{2\pi {{I}_{2}}}{{{r}_{2}}}=\frac{{{\mu }_{0}}{{I}_{2}}}{2{{r}_{2}}}\] Since, the coil \[{{C}_{1}}\] is small, \[{{B}_{2}}\]remains constant, Therefore, magnetic flux linked with the coils \[{{C}_{1}},\] \[{{\phi }_{12}}={{B}_{2}}\times \pi r_{1}^{2}=\frac{{{\mu }_{0}}{{I}_{2}}}{2{{r}_{2}}}\times \pi r_{1}^{2}\]            ..? (i) If M is coefficient of mutual inductance between the two coils, then \[{{\phi }_{12}}=M{{I}_{2}}\]  ?..?.. (ii) From Eqs. (i) and (ii), we get, \[M=\frac{{{\mu }_{0}}\pi r_{1}^{2}}{2{{r}_{2}}}\] (ii) Let coil rotates by an angle q with the magnetic field in time t. \[\omega =\frac{\theta }{t}\] \[\Rightarrow \] \[\theta =\omega \,t\]  ???. (i) \[\therefore \] Magnetic flux linked with each turn of rectangular coil, \[\phi =BA\,cos\,\theta =BA\,cos\,\omega t\] \[d\phi \,/\,dt=-BA\,\omega \,sin\,\omega t\] Where,              \[\omega =2\pi f\] \[\Rightarrow \] For N turn, \[-N\,(d\phi \,/\,dt)=BNA\,\omega \,sin\,\omega t\] ??? (ii) By Faraday?s law,           \[e=-N(d\phi \,/\,dt)\] ..?.? (iii) From Eqs. (ii) and (iii), we get Induced emf, \[e=NBA\,\omega \,\,sin\,\omega t\] \[\omega =2\pi f,\,\,e=NBA(2\pi f)\,sin\,\omega t\] \[e={{e}_{0}}\,sin\,\omega t\]    ??..? (iv) Where, e = maximum value of emf induced in coil and given by, \[{{e}_{0}}=2\pi f\,\,NBA\] Or Mutual inductance The mutual inductance of a pair of coils, equals the magnetic flux linked with one of them due to a unit current in the other. Factors affecting the mutual inductance of a pair of coils (i) The sizes of the two coils. (ii) The shape of the two coils. (iii) The distance of separation between the two coils. (iv) The nature of the medium between the two coils. (v) The relative orientation of the two coils from \[t=0\] to \[t=3s=\left( \frac{30\,\,cm}{10\,\,cm/s} \right),\] the flux through the coil is zero. From \[t=3s\]to \[\grave{\ }t=5s,\] the flux through the coil remains zero. (i) The plot of \[\phi \]against \[t\] is as given below (ii) \[E=-(d\phi \,/\,dt),\]the plot of E against t is as given below (iii) \[i=\frac{E}{R}=\frac{2\times {{10}^{-3}}}{0.1\,\Omega }=20\,\,mA\]