Question

A plane circular coil is rotated about its vertical diameter with a constant angular speed \(\omega\) in a uniform horizontal magnetic field. Initially the plane of the coil is parallel to the magnetic field. Draw the plot showing the variation of magnetic flux \(\phi\) linked with the coil as a function of \(\omega t\), where \(t\) represents the time elapsed.

Hint:

If the plane of the coil is initially parallel to the magnetic field, the initial flux is zero. Therefore the flux graph starts from the origin and follows a sine curve: \[ \phi=\phi_0\sin(\omega t). \]

Answer 1

Concept: The magnetic flux linked with a coil placed in a magnetic field is given by \[ \phi = BA\cos\theta, \] where
• \(B\) is the magnetic field strength,
• \(A\) is the area of the coil,
• \(\theta\) is the angle between the magnetic field and the normal to the plane of the coil. When the coil rotates with a constant angular speed \(\omega\), the angle between the normal to the coil and the magnetic field changes continuously with time. Hence, the magnetic flux varies sinusoidally.

Step 1: Determine the initial orientation of the coil. Initially, the plane of the coil is parallel to the magnetic field. Therefore, the normal to the coil is perpendicular to the magnetic field. Hence, \[ \theta = 90^\circ. \] Therefore, the initial magnetic flux is \[ \phi = BA\cos 90^\circ = 0. \] Thus, at \[ t=0, \] \[ \phi =0. \]

Step 2: Write the angle as a function of time. As the coil rotates with angular speed \(\omega\), \[ \theta = 90^\circ-\omega t. \] Therefore, \[ \phi = BA\cos(90^\circ-\omega t). \] Using \[ \cos(90^\circ-x)=\sin x, \] we obtain \[ \phi = BA\sin(\omega t). \] Let \[ \phi_0=BA. \] Hence, \[ \boxed{\phi=\phi_0\sin(\omega t)}. \]

Step 3: Determine important points of the graph. At \[ \omega t=0, \] \[ \phi=0. \] At \[ \omega t=\frac{\pi}{2}, \] \[ \phi=\phi_0. \] At \[ \omega t=\pi, \] \[ \phi=0. \] At \[ \omega t=\frac{3\pi}{2}, \] \[ \phi=-\phi_0. \] At \[ \omega t=2\pi, \] \[ \phi=0. \] Thus the graph is a sine curve starting from zero and initially increasing in the positive direction. Required Plot: \[ \phi=\phi_0\sin(\omega t) \] \[ \begin{array}{c} \text{Magnetic Flux }(\phi) \phi_0 \quad\quad\quad\quad\quad \bullet \quad\quad\quad\quad / \backslash \quad\quad\quad / \quad \backslash 0 \bullet\quad\quad\quad\quad\bullet\quad\quad\quad\quad\bullet \quad\quad\quad \backslash \quad / \quad\quad\quad\quad \backslash / -\phi_0 \quad\quad\quad\quad\bullet \end{array} \] \[ 0 \qquad \frac{\pi}{2} \qquad \pi \qquad \frac{3\pi}{2} \qquad 2\pi \] along the \(\omega t\)-axis.