Question
A rectangular metallic loop of length l and width b is placed coplanarly with a long wire carrying a current i. The loop is moved perpendicular to the wire with a speed v in the plane containing the wire and the loop. Calculate the emf induced in the loop when the rear end of the loop is at a distance a from the wire. Solve by using Faraday's law for the flux through the loop and also by replacing different segments with equivalent batteries.

Answer

Using Faraday’' law

Consider a unit length dx at a distance x

$\text{B}=\frac{\mu_0\text{i}}{2\pi\text{x}}$

Area of strip $=\text{b} \ \text{dx}$

$\text{d}\phi=\frac{\mu_0\text{i}}{2\pi\text{x}}\text{dx}$

$\Rightarrow\phi=\int\limits^{\text{a}+1}_\text{a}\frac{\mu_0\text{i}}{2\pi\text{x}}\text{bdx}$

$=\frac{\mu_o\text{i}}{2\pi}\text{b}\int\limits^{\text{a}+1}_\text{a}\Big(\frac{\text{dx}}{\text{x}}\Big)=\frac{\mu_0\text{ib}}{2\pi}\log\Big(\frac{\text{a}+\text{l}}{\text{a}}\Big)$

$\text{Emf}=\frac{\text{d}\phi}{\text{dt}}=\text{dt}\Big[\frac{\mu_0\text{ib}}{2\pi}\text{log}\Big(\frac{\text{a}+\text{l}}{\text{a}}\Big)\Big]$

$=\frac{\mu_0\text{ib}}{2\pi}\frac{\text{a}}{\text{a}+\text{l}}\Big(\frac{\text{va}-(\text{a}+\text{l})\text{v}}{\text{a}^2}\Big)$ $\Big($ Where $\frac{\text{da}}{\text{dt}}=\text{V}\Big)$

$=\frac{\mu_0\text{ib}}{2\pi}\frac{\text{a}}{\text{a}+\text{l}}\frac{\text{vl}}{\text{a}^2}=\frac{\mu_0\text{ibvl}}{2\pi(\text{a}+\text{l})\text{a}}$

The velocity of AB and CD creates the emf. since the emf due to AD and BC are equal and opposite to each other.

$\text{B}_{\text{AB}}=\frac{\mu_o\text{i}}{2\pi\text{a}} \ \Rightarrow \ \text{E.m.f.} \ \text{AB}=\frac{\mu_0\text{i}}{2\pi\text{a}}\text{bv}$

Length b, velocity v.

$\text{B}_{\text{CD}}=\frac{\mu_0\text{i}}{2\pi(\text{a}+\text{l})}$

$\Rightarrow \text{E.m.f.} \ \text{CD}=\frac{\mu_0\text{ibv}}{2\pi(\text{a}+\text{l})}$

Length b, velocity v.

Net emf $=\frac{\mu_0\text{i}}{2\pi\text{a}}\text{bv}-\frac{\mu_0\text{ibv}}{2\pi\text{a}(\text{a}+\text{l})}=\frac{\mu_0\text{ibvl}}{2\pi\text{a}(\text{a}+\text{l})}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Similar questions

A voltmeter of resistance $400\Omega$ is used to measure the potential difference across the $100\Omega$ resistor in the circuit shown in the figure. (a) What will be the reading of the voltmeter? (b) What was the potential difference across $100\Omega$ before the voltmeter was connected?

A compound microscope consists of an objective lens of focal length 2.0 cm and an eyepiece of focal length 6.25 cm separated by a distance of 15 cm. How far from the objective should an object be placed in order to obtain the final image at (a) the least distance of distinct vision (25 cm), and (b) at infinity? What is the magnifying power of the microscope in each case?
An air bubble of radius 2.0mm is formed at the bottom of a 3.3m deep river. Calculate the radius of the bubble as it comes to the surface. Atmospheric pressure = 1.0 × 105 Pa and density of water = 1000kg/m-3.
What do you understand by a solenoid? Obtain an expression for magnetic field of a solenoid of infinite length.###Establish a formula for magnetic field on the axis inside of a straight and long solenoid. Show the direction of magnetic field inside it and what is the magnitude of magnetic field outside the solenoid.###Establish a formula for magnetic field inside a long solenoid. Represent produced force lines by diagram.###Write Ampere's circuital law. Obtain an expression for magnetic field on the axis of a very long current carrying solenoid. Draw necessary diagram also.
Calculate the increase in the internal energy of 10g of water when it is heated from 0°C to 100°C and converted into steam at 100kPa. The density of steam= 0.6kg/ m-3. Specific heat capacity of water = 4200J/ kg-1°C-1 and the Jatent heat of vaporization of water = 2.25 × 106J/kg-1.

AC = CO = D, S1C = S2C = d < < D

A small transparent slab containing material of μ = 1.5 is placed along AS2 (Fig). What will be the distance from O of the principal maxima and of the first minima on either side of the principal maxima obtained in the absence of the glass slab.

A bulb with rating 250V, 100W is connected to a power supply of 220V situated 10m away using a copper wire of area of cross-section 5mm2. How much power will be consumed by the connecting wires? Resistivity of copper $1.7\times10^{-8}\Omega-\text{m}$.
Air is pumped into the tubes of a cycle rickshaw at a pressure of 2 atm. The volume of each tube at this pressure is 0.002m3. One of the tubes gets punctured and the volume of the tube reduces to 0.0005m3. How many moles of air have leaked out? Assume that the temperature remains constant at 300K and that the air behaves as an ideal gas.
Define induction coefficient. Explain the phenomenon of mutual inductance with the help of an experiment. Obtain an expression for mutual inductance coefficient of two solenoids and prove that $M=\sqrt{L_1 L_2}$.
State the Biot-Savart law for the magnetic field due to a current carrying element. Use this law to obtain a formula for magnetic field at the centre of a circular loop of radius R carrying a steady current I. Sketch the magnetic field lines for a current loop clearly indicating the direction of the field.