Advertisements
Advertisements
Question
Consider an infinitely long wire carrying a current I(t), with `(dI)/(dt) = λ` = constant. Find the current produced in the rectangular loop of wire ABCD if its resistance is R (Figure).
Advertisements
Solution
To approach these types of problems integration is very useful to find the total magnetic flux linked with the loop.
Let us first consider an elementary strip of length l and width dr at a distance r from an infinite long current carrying wire. The magnetic field at strip due to current carrying wire is given by
`l = B(r) = (mu_0I)/(2pir) l.B(r)`
∴ Flux in strip `phi = (mu_0I)/(2pi) l int_(x_0)^x (dr)/r`
`phi = (mu_0Il)/(2pi) [log_e r]_(x_0)^x = (mu_0Il)/(2pi) log_e x/x_0`
`ε = (-dphi)/(dt)`
So `IR = (dphi)/(dt)`
`I = 1/R d/(dt) [(mu_0Il)/(2pi) log_e x/x_0] = (mu_0l)/(2piR) . log_e x/x_0 (dl)/(dt)`
`I = (mu_0λl)/(2piR) log_e x/x_0` .....`[because (dl)/(dt) = λ (given)]`
APPEARS IN
RELATED QUESTIONS
Current in a circuit falls from 5.0 A to 0.0 A in 0.1 s. If an average emf of 200 V induced, give an estimate of the self-inductance of the circuit.
The currents flowing in the two coils of self-inductance L1 = 16 mH and L2 = 12 mH are increasing at the same rate. If the power supplied to the two coil is equal, find the ratio of the currents ?
Define self-inductance of a coil. Show that magnetic energy required to build up the current I in a coil of self inductance L is given by `1/2 LI^2`
Derive the expression for the self-inductance of a long solenoid of cross sectional area A and length l, having n turns per unit length.
Derive an expression for self inductance of a long solenoid of length l, cross-sectional area A having N number of turns.
Consider the self-inductance per unit length of a solenoid at its centre and that near its ends. Which of the two is greater?
A constant current i is maintained in a solenoid. Which of the following quantities will increase if an iron rod is inserted in the solenoid along its axis?
(a) magnetic field at the centre
(b) magnetic flux linked with the solenoid
(c) self-inductance of the solenoid
(d) rate of Joule heating.
Two solenoids have identical geometrical construction but one is made of thick wire and the other of thin wire. Which of the following quantities are different for the two solenoids?
(a) self-inductance
(b) rate of Joule heating if the same current goes through them
(c) magnetic field energy if the same current goes through them
(d) time constant if one solenoid is connected to one battery and the other is connected to another battery.
An inductor-coil carries a steady-state current of 2.0 A when connected across an ideal battery of emf 4.0 V. If its inductance is 1.0 H, find the time constant of the circuit.
A coil having inductance 2.0 H and resistance 20 Ω is connected to a battery of emf 4.0 V. Find (a) the current at the instant 0.20 s after the connection is made and (b) the magnetic field energy at this instant.
Answer the following question.
When an inductor is connected to a 200 V dc voltage, a current at 1A flows through it. When the same inductor is connected to a 200 V, 50 Hz ac source, only 0.5 A current flows. Explain, why? Also, calculate the self-inductance of the inductor.
When a coil is connected to a D.C. source of e.m.f. 12 volt, then a current of 4 ampere flows in it. If the same coil is connected to a 12 volt, 25 cycle/s A.C sources, then the current flowing in it is 2.4 A. The self-inductance of the coil will be ______
Consider a solenoid carrying supplied by a source with a constant emf containing iron core inside it. When the core is pulled out of the solenoid, the change in current will ______.
A graph of magnetic flux `phi` versus current (I) is shown for four inductors A, B, C, D. Smaller value of self inductance is for inductor ____________.
When the current in a coil changes from 2 amp. to 4 amp. in 0.05 sec., an e.m.f. of 8 volt is induced in the coil. The coefficient of self inductance of the coil is ______.
When current i passes through an inductor of self-inductance L, energy stored in it is 1/2. L i2. This is stored in the ______.
The inductance of a solenoid L having diameter d. Let n be the number of turns per unit length. The inductance per unit length near the middle of a solenoid is ______.
(Assume that, l = current passes through the turns, µ0 = Permeability of vacuum)
In a coil, the current changes form −2 A to +2 A in 0.2 s and induces an emf of 0.1 V. The selfinductance of the coil is ______.
What does \[I\] represent, and what is its SI unit?
After whom is the SI unit henry (H) named?
