Q24 of 26 Page 1

(a) Define self-inductance of a coil. Obtain an expression for the energy stored in a solenoid of self-inductance ‘L’ when the current through it grows from zero to ‘I’.

(B) Derive the expression for the magnetic field due to a solenoid of length ‘2 l’, radius ‘a’ having ‘n’ number of turns per unit length and carrying a steady current ‘I’ at a point on the axial line, distant ‘r’ from the centre of the solenoid. How does this expression compare with the axial magnetic field due to a bar magnet of magnetic moment ‘m’


OR


(a) Draw the magnetic field lines due to a circular loop of area Acarrying current I. Show that it acts as a bar magnet of magnetic momentum AI.


(b) Derive the expression for the magnetic field due to a solenoid of length ‘2 l’, radius ‘a’ having ‘n’ number of turns per unit length and carrying a steady current ‘I’ at a point on the axial line, distant ‘r’ from the centre of the solenoid. How does this expression compare with the axial magnetic field due to a bar magnet of magnetic moment ‘m’ ?


(a) Self-inductance is defined as induced emf produced per unit rate change of current through the inductance coil i.e.



When the current is changing, then the flux associated with the coil changes and an e.m.f. is induced in the coil. It is given by



The self-induced e.m.f. is also called back e.m.f. as it opposes any change in current in the circuit. Therefore, work should be done against the back e.m.f. in establishing current. This work done is stored as magnetic potential energy.


The rate of doing work is given as



Thus, the total work done in inducing current from 0 to I is



(b) The direction of induced current in the loop as it goes out is depicted in the figure below



The current will persist till the entire loop comes out of the field. Hence, we have



Hence, the current will persist for 1 second.


(ii) The magnetic flux in the coil when it is inside the field is constant. The maximum flux is given as ϕ= Bla where a is the side of the square loop. This flux will start decreasing once the loop starts coming out of the field and will be zero when it is completely out of the field. The e.m.f. induced in the coil when it is inside the field is zero because the flux is not changing. When the loop just comes out of the field, the flux change is maximum and the e.m.f. induced is given as



This e.m.f remains constant till the entire loop comes out. When the loop is completely out of the field, the e.m.f. drops to zero again.



OR


(a) When a circular loop carries current I with the area A, the loop develops a magnetic field around itself. The strength of the magnetic field developed depends on the current running through the wire. At the centre of the loop, the magnetic field lines are perpendicular to the plane of the loop which gives the magnetic moment, m = IA. Hence, it behaves like a magnet.


When we compare the magnetic field lines around a current-carrying loop and a bar magnet, it shows that both the patterns are similar. If the current in the loop is in the anticlockwise direction, a North Pole is formed and if the current is in the clockwise direction, a South Pole is formed



(b) Consider a solenoid of length 2l, radius a and having n turns per unit length. It is carrying a current I. We have to evaluate the axial field at a point P at a distance r from the centre of the solenoid.



Let us consider the small element of thickness dx of the solenoid as shown in the figure. Let it be x distance away from its centre O. the magnetic field due to the small circular loop carrying current I is given as



Integrating both the sides



For r>>a and r>>1, the , hence,



The magnitude of the magnetic moment is given as the product of total number of turns of wire, current lowing in the wire and cross section area of the solenoid. Therefore,



Therefore, the magnetic field is given as



This is the magnetic field due to a solenoid on the axial line at a distance r from the centre.


More from this chapter

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22

Explain briefly the process of charging a parallel plate capacitor when it is connected across a d.c. battery.

A capacitor of capacitance ‘C’ is charged to ‘V’ volts by a battery. After some time, the battery is disconnected and the distance between the plates is doubled. Now a slab of dielectric constant, 1 < k < 2, is introduced to fill the space between the plates. How will the following be affected: (a) The electric field between the plates of the capacitor


(b) The energy stored in the capacitor


Justify your answer by writing the necessary expressions.


 23

Ajit had a high-tension tower erected on his farm land. He kept complaining to the authorities to remove it as it was occupying a large portion of his land. His uncle, who was a teacher, explained to him the need for erecting these towers for efficient transmission of power. As Ajit realized its significance, he stopped complaining. Answer the following questions:

(a) Why is it necessary to transport power at high voltage?


(b) A low power factor implies large power loss. Explain.


(c) Write two values each displayed by Ajit and his uncle.


 25

(a) In Young’s double slit experiment, deduce the conditions for obtaining constructive and destructive interference fringes. Hence deduce the expression for the fringe width.

(b) Show that the fringe pattern on the screen is actually a superposition of single slit diffraction from each slit.


(c) What should be the width of each slit to obtain 10 maxima of the double slit pattern within the central maximum of the single slit pattern, for green light of wavelength 500 nm, if the separation between two slits is 1 mm ?


OR


(a) Two thin convex lenses L1 and L2 of focal lengths f1 and f2 respectively, are placed coaxially in contact. An object is placed at a point beyond the focus of lens L1. Draw a ray diagram to show the image formation by the combination and hence derive the expression for the focal length of the combined system.


(b) A ray PQ incident on the face AB of a prism ABC, as shown in the figure, emerges from the face AC such that AQ = AR. Draw the ray diagram showing the passage of the ray through the prism. If the angle of the prism is 60°and refractive index of the material of the prism is 3, determine the values of angle of incidence and angle of deviation.


 26

(a) Deduce the expression for the potential energy of a system of two charges q1 and q2 located at r and r respectively in an external electric field.

(b) Three points charges, + Q, + 2Q and – 3Q are placed at the vertices of an equilateral triangle ABC of side l. If these charges are displaced to the mid-points A1, B1 and C1 respectively, find the amount of the work done in shifting the charges to the new locations.


OR


Define electric flux. Write its S.I. unit. State and explain Gauss’s law. Find out the outward flux due to a point charge + q placed at the centre of a cube of side ‘a’. Why is it found to be independent of the size and shape of the surface enclosing it? Explain.