Title Med-RM_Phy_SP-4_Ch-21_Electromagnetic Induction.pdf ✅

Aakash Educational Services Limited - Regd. Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 Ph. 011-47623456 Chapter Contents Chapter 21 Electromagnetic Induction Introduction Magnetic Flux Faraday’s Law of Induction Motional Electromotive Force Induced Electric Field Energy Consideration : A Quantitative Study Eddy Current Self-Inductance Self Inductances of Solenoid Mutual Inductance L-R Circuit AC Generator D.C Motor Introduction Scientist Oersted demonstrated by his experiments that electric currents can produce (induce) magnetic field in space around it. The reverse of results of Oersted’s experiments, was proved by the experimental demonstration of Faraday. The conclusion of these experiments was the electric current is generated by varying magnetic field. In this chapter, we will study the experiments of electromagnetic induction and their applications. MAGNETIC FLUX It is proportional to the number of magnetic lines passing through a surface. Mathematically . cos    B B A BA ...(i) Where B is the magnetic flux through a plane surface of area A placed in a uniform magnetic field B.  is the angle between and . B A Equation (i) can be extended to curved surfaces and non-uniform fields. B A  If the magnetic field has different magnitudes and directions at various parts of a surface as shown in figure, then the magnetic flux through the surface is given by
86 Electromagnetic Induction NEET Aakash Educational Services Limited - Regd. Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 Ph. 011-47623456 dAi Bi Fig.: Magnetic field at the area element. represents area vector of the area element. B i dA i i i th th 1 2 . . ... . 1 2 i n B i i i l B dA B dA B dA        ...(ii) Unit of magnetic flux weber Dimension of magnetic flux [ML2 T–2 A–1] Example 1 : A uniform magnetic field exists in the space -12 3 B Bi B j B k   . Find the magnetic flux through an area S if the area S is in (i) x-y plane (ii) y-z plane (iii) z-x plane Solution : (i) Since the field is uniform, we can use formula .   B B S Now when area S is in x-y plane, it means - S Sk Hence --12 3 3 ( ).( )      B B i B j B k Sk B S --Example 2 : Figure shows a long straight wire carrying current I and a square conducting wire loop of side l, at a distance ‘a’ from current wire. Both the current wire and loop are in the plane of paper. Find the flux of magnetic field of current wire, passing through the loop. x x x x x x x + – x dx a I Solution : Since the field of current wire passing through the loop, is same in direction (normally inward) but not uniform in magnitude. Hence we will use integration method for finding the flux. The small flux through a thin rectangular strip of length l and width dx, is given by 0 ( ). ( ) cos180 . 2 B I d B x dA B x dA dx x             0 0 . ,..log 2 2 xal BB e x a I d dx I x x               = – 0 . log 2    e a l I a
NEET Electromagnetic Induction 87 Aakash Educational Services Limited - Regd. Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 Ph. 011-47623456 FARADAY’S LAW OF INDUCTION On the basis of experiments faraday gave two laws about electromagnetic induction. (1) If there is a change in magnetic flux linked with a coil then an e.m.f induced in the coil. (2) The magnitude of the induced emf in a circuit is equal to the time rate of change of magnetic flux through the circuit. Mathematically, Instantaneous emf () () d et t dt   The average induced emf 2 1 2 1              e t tt The negative sign indicates the direction of induced emf and hence the direction of induced current in the loop opposes cause of its generation which is in accordance with Lenz's law. In case of a closely wound coil of N turns, induced emf () () d et N t dt   Lenz’s Law The direction of induced emf (i.e., polarity of induced emf) and hence the direction of induced current in a closed circuit is to oppose the cause due to which they are produced. For example, if the flux is increasing, induced emf (and hence induced current) will try to decrease the flux and vice-versa. It is based on energy conservation law. Methods to Change the Magnetic Flux Here are the general methods by which we can change the magnetic flux through a coil. (i) Change the magnitude B of the magnetic field within the coil. (ii) Change either the total area of the coil or the portion of that area that lies within the magnetic field (for example, by expanding the coil or sliding it in or out of the field. (iii) Change the angle between the direction of the magnetic field B and the plane of coil (For example, by rotating the coil so that field B is first perpendicular to the plane of the coil and then is along that plane). Note : (i) Changing a magnetic field in a loop induces voltage. If the loop is an electrical conductor forming a closed circuit, then current is induced (ii) Induced current () 1 () . () et d it t R R dt    (iii) Induced charge 1 1 ( ) | | d q dq i t dt dt d R dt R          Hence 1    q | | R q is independent of time interval during which flux is changing. Applications of Lenz's law shall become more clear by carefully studying the following examples (1) A conducting loop is kept in a uniform magnetic field B directed into the plane of paper. The magnetic field starts increasing with time. As the magnetic field increases, flux  AB ).( starts increasing. Therefore, an induced current will flow into the loop. The current is induced due to increasing field, therefore, it will