91影视

i) Radius is, r=10 cm

ii) Rate of increase of magnetic field is,$\frac{\mathrm{dB}}{\mathrm{dt}}=3脳{10}^{-3}\mathrm{T}/\mathrm{s}$

First, find the expression for magnetic flux and then substitute it in the equation for Faraday鈥檚 law. By substituting the given values, get the emf developed in the wire. Using Lenz鈥檚 law, find the direction of the induced current in the wire segment.

Faraday'slaw of electromagnetic inductionstates, Whenever a conductor is placed in a varying magnetic field, an electromotive force is induced in it.

Lenz's law states that the current induced in a circuit due to a change in a magnetic field is directed to oppose the change in flux and to exert a mechanical force which opposes the motion.

Formulae are as follow:

$$\begin{gathered} 蠒=鈭�\stackrel{鈫�}{B}.d\stackrel{鈫�}{A} \\ 蔚=-\frac{d蠒}{dt} \end{gathered}$$

Where,$桅$is magnetic flux, B is a magnetic field, A is an area,饾渶 is emf.

Themagnetic field is along +x axis, i.e.,

$\stackrel{鈫�}{B}=B\stackrel{铃�}{i}$

So the magnetic flux only passes through the area on yz plane.

The area on the yz plane is one-fourth of the area ofthecircle.

The magnetic flux through the loop is given by,

$$\begin{gathered} {蠒}_B=鈭�\stackrel{鈫�}{B}.d\stackrel{鈫�}{A} \\ {蠒}_B=鈭�BdA\cos 胃 \end{gathered}$$

Where,$胃$is the angle betweenthemagnetic field vector and the area vector.

Asthemagnetic field and area vector is in the same direction,$\cos 胃=\cos 0=1$

Since the magnetic field is uniform, it is taken out of the integral.

${蠒}_B=B鈭�dA$

$鈭�dA=\frac{1}{4}蟺r^2$

${蠒}_B=\frac{1}{4}蟺r^{2}B$

The magnitude of emf is given by Faraday鈥檚 law.

$\left|蔚\right|=\left|\frac{d{蠒}_B}{dt}\right|$

Thus,

$$\begin{gathered} \left|\mathrm{蔚}\right|=\left|\frac{\mathrm{d}}{\mathrm{dt}}\left(\frac{1}{4}{\mathrm{蟿蟿谤}}^2\mathrm{B}\right)\right|=\frac{{\mathrm{蟿蟿谤}}^2}{4}\left|\frac{\mathrm{dB}}{\mathrm{dt}}\right| \\ \left|\mathrm{蔚}\right|=\frac{1}{4}\mathrm{蟿蟿}{\left(0.10\mathrm{m}\right)}^2\left(3脳{10}^{-3}\frac{\mathrm{T}}{\mathrm{s}}\right) \\ \left|\mathrm{蔚}\right|=2.4脳{10}^5\mathrm{V} \end{gathered}$$

Hence, magnitude of the emf developed in the wire when increases at the rate of 3mT/s is,$2.4脳{10}^{-5}\mathrm{V}$

Since, the magnetic field is along positive x axis, the induced field is in its opposite direction. Thus by Lenz鈥檚 law, the current in the coil must flow from point c to point b

Hence, the direction of the current in segmentbcis from c tob

Therefore, find the magnetic flux by using Faraday鈥檚 law and the direction of the current by using Lenz鈥檚 law.