91影视

i) Diameter of the loop$d_L=10cm=0.10m$

ii) Diameter of the wire$d_w=2.5mm=00.0025m$

iii) Force acting on the particle $蚁=1.69脳{10}^{-8}惟-m$

iv) Induced current i = 10 A$i=10A$

By using expression for Magnetic flux, Faraday鈥檚 law, Resistivity, Ohm鈥檚 law, find the rate of change of the magnetic field.

Faraday's law of electromagnetic induction states,Whenever a conductor is placed in a varying magnetic field, an electromotive force is induced in it.

Ohm's law states that the voltage across a conductor is directly proportional to the current flowing through it, provided all physical conditions and temperatures remain constant.

Formulae are as follow:

i) Magnetic flux,$蠁=鈭�BdA$

ii) Faraday鈥檚 law,$蔚=-\frac{d蠁}{dt}$

iii) Resistance,$d=\frac{蚁L}{A}$

iv) Ohm鈥檚 law ,$V=IR$

Where,$桅$is magnetic flux, B is magnetic field, A is area, I is current, R is resistance,饾渶 is emf, L is length, V IS voltage,饾湆 is resistivity.

Magnetic flux through the loop area is given by,

$$\begin{gathered} 蠁=鈭�Bd{A}_L \\ 蠁=B{A}_L \end{gathered}$$

Induced emf loop in the loop is,

$$\begin{gathered} 蔚=\frac{d蠁}{dt} \\ 蔚=\frac{d\left(B{A}_L\right)}{dt} \\ \frac{蔚}{A_L}=\frac{dB}{dt}........................................................\left(1\right) \end{gathered}$$

Area of the loop $A_L$is given by,

$A_L=\frac{蟿蟿d_L^2}{4}.............................................................\left(2\right)$

By Ohm鈥檚 law,

$蔚=IR................................................................\left(3\right)$

To find the resistance of the wire,

$R=\frac{蚁L}{A_W}$

Length of the wire L= circumference of the loop =$蟿蟿d_L$

Area of the cross section of the wire is given by,$A_W=\frac{蟿蟿d_W^2}{4}$

Therefore, the resistance of the wire is,

$$\begin{gathered} R=蚁\frac{蟿蟿d_L}{{\displaystyle \frac{蟿蟿d_W^2}{4}}} \\ R=\frac{4蚁d_L}{d_W^2} \end{gathered}$$

Using in equation 3,

$$\begin{gathered} 蔚=IR \\ 蔚=I脳\frac{4蚁d_L}{d_W^2}.........................................................\left(4\right) \end{gathered}$$

Using equations 2 and 4 in 1,

$$\begin{gathered} \frac{蔚}{A_L}=\frac{dB}{dt} \\ \frac{dB}{dt}=I脳\frac{4蚁d_L}{d_W^2}脳\frac{4}{蟿蟿d_L^2} \\ \frac{dB}{dt}=\frac{16Ip}{蟿蟿d_L{d}_W^2} \end{gathered}$$

Using all the given values,

$$\begin{gathered} \frac{dB}{dt}=\frac{16脳10脳1.69脳{10}^{-8}}{3.14脳0.10脳0.{0025}^2} \\ \frac{dB}{dt}=1.37T/s \\ \frac{dB}{dt}=1.4T/s \end{gathered}$$

Hence, the rate of change of the magnetic field is,$\frac{dB}{dt}=1.4\frac{T}{s}$ .

Therefore, by using expression for Magnetic flux, Faraday鈥檚 law, Resistivity, Ohm鈥檚 law, the rate of change of the magnetic field can be found.