First slide

Magnetic flux and induced emf

Question

A copper wire of length 2 m placed perpendicular to the plane of magnetic field $\vec{B} = (2 \hat{i} + 4 \hat{j})$ T.If it moves with velocity $(4 \hat{i} + 6 \hat{j} + 8 \hat{k})$ m/sec. The magnitude of dynamic emf (in volt) across its ends is __________.

Moderate

Solution

$\begin{array}{l} Dynamic emf e_{d} = \vec{l} . (\vec{B} \times \vec{v}) \\ \begin{matrix} (\vec{B} \times \vec{v}) = |\begin{array}{ll} \hat{i} & \hat{j} & \hat{k} \\ 2 & 4 & 0 \\ 4 & 6 & 8 \end{array}| \\ = \hat{i} (32 - 0) - \hat{j} (16 - 0) + \hat{k} (12 - 16) = 32 \hat{i} - 16 \hat{j} - 4 \hat{k}, \\ e_{d} = (2 \hat{k}) \cdot (32 \hat{i} - 16 \hat{j} - 4 \hat{k}) = - 8 volt \end{matrix} \end{array}$

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