question_answer

A large circular coil of radius R and a small circular coil radius r are put in vicinity of each other. If the coefficient of mutual induction for this pair, equals 1 mH, what would be the flux linked with the larger coil when a current of 0.5 A flows through the smaller coil?

When the current in the smaller coil falls to zero, what would be its effect in the larger coil?

Or

Show that the total induced charge simply depends upon the change in the magnetic flux and is independent of the time rate of change of flux.

Answer:

\[5\times {{10}^{-\,4}}Wb\] According to the question, Mutual inductance for given power of coil,             \[M=1\text{mH}=1\times {{10}^{-3}}\text{H}\]             \[l=0.5\,\text{A}\] \[\therefore \] Total magnetic flux linked with larger coil,                         \[\phi =MI\]                         \[\phi =(1\times {{10}^{-3}})\times 0.5\]                         \[\phi =5\times {{10}^{-4}}Wb\] With the fall of current in small coil to zero, the magnetic flux linked with bigger coil decrease to zero which produces an induced emf in it.                                     Or According to the Faraday?s law, the induced emf \[(\varepsilon )\] in the circuit is given by, \[\varepsilon =\frac{-\,d{{\phi }_{B}}}{dt}\] Where, \[{{\phi }_{B}}\] is the magnetic flux. If R be the resistance of the circuit, then the induced current is given as                         \[I=\frac{\varepsilon }{R}=\frac{-1}{R}\frac{d{{\phi }_{B}}}{dt}\]                                  ? (i) Also,     \[I=\frac{dQ}{dt}\] \[\Rightarrow \]   \[dQ=I\cdot dt\] Substituting the value of I from Eq. (i), we get             \[dQ=\frac{-1}{R}\frac{d{{\phi }_{B}}}{dt}\cdot dt=-\frac{1}{R}d{{\phi }_{B}}\] \[\therefore \] The total induced charge                         \[Q=\frac{-1}{R}\int{d{{\phi }_{B}}.=\frac{-1}{R}[{{\phi }_{\text{f}}}-{{\phi }_{i}}]}\] \[=\frac{1}{R}[{{\phi }_{i}}-{{\phi }_{\text{f}}}]\]                                 ? (ii) Where, \[{{\phi }_{i}}=\] initial value of the magnetic Flux and \[{{\phi }_{\text{f}}}=\] final value of the magnetic flux. Hence, it is clear from the above Eq. (ii) that total induced charge depends upon the change in the magnetic flux and is independent of the time rate of change of flux.