Chapter 5: Electromagnetic Induction 5.1 Magnetic Flux 5.1.1 Define and use magnetic flux Magnetic flux is defined as the scalar product between the magnetic flux density, B with the vector of the area, A. It is a measure of the number of magnetic field lines that cross a given area. Mathematically, B A BAcos It is a scalar quantity. Unit SI for Φ is T m or Wb ( weber ) where Ф is magnetic flux θ is the angle between and If the coil is composed of N turns, all of the same area A, thus the magnetic flux through N turns coil (magnetic flux linkage) is: Note: NBAcos Direction of vector A always perpendicular (normal) to the surface area, A. The magnetic flux is proportional to the number of field lines passing through the area. Example A single turn of rectangular coil of sides 10 cm 5.0 cm is placed between north and south poles of a permanent magnet. Initially, the plane of the coil is parallel to the magnetic field as shown in Figure. Solution If the coil is turned by 90 about its rotation axis and the magnitude of magnetic flux density is 1.5T, calculate the change in the magnetic flux through coil. BP3 1 FYSL
5. Induced emf Electromagnetic induction is the production of an induced e.m.f. (or voltage) across a conductor or circuit situated in a changing magnetic field. The meaning of changing in magnetic flux: There is a relative motion of loop & magnet field lines are cut : The number of magnetic field lines passing through a coil are increased or decreased: 5..1 Use Faraday s experiment to explain induced emf Faraday s Experiment When there is no relative motion between the magnet & the loop, G shows no deflection. No induced current. Moving the magnet toward the loop increases the number of magnetic field lines passing through loop. The G needle is deflected indicating an induced current is produced. Moving the magnet away from the loop decreases the number of magnetic field lines passing through the loop. The induced current is now in opposite direction. Conclusion: From the experiments, it can be seen that e.m.f is induced only when the magnetic flux through the coil change. BP3 FYSL
The magnitude of induced e.m.f. can be increased by: Increasing the number of turns, N Increasing the strength of magnet/ use stronger magnet (magnetic flux increased), B Increasing the area of the coil or solenoid, A Move the magnet into or out the solenoid faster 5.. State Faraday s law and use Lenz s law to determine the direction of induced current Faraday s law of induction states that the magnitude of the induced e.m.f. is proportional to the rate of change of the magnetic flux. Mathematically, d where dф is change of magnetic flux is change of time The negative sign indicates that the direction of induced e.m.f. always oppose the change of magnetic flux producing it (Lenz s law). (To calculate the magnitude of induced e.m.f., the negative sign can be ignored.) Lenz s law states that an induced current always flow in a direction that opposes the change in magnetic flux that causes it. By Lenz s Law, when magnet is inserted into the solenoid, a North pole will be induced on the right side of coil to oppose the incoming North pole. By right hand grip rule, the induced current will flow anticlockwise so that pointer deflects to right. By Lenz s Law, when magnet is withdrawn from the solenoid, a South pole will be induced on the right side of coil to oppose the outgoing North pole. By right hand grip rule, the induced current will flow clockwise so that pointer deflects to left Lenz s Law is an example of principle of Conservation of Energy. Mechanical work is done to against the opposing magnetic force experienced by the moving magnet, and this work is converted into electrical energy as indicated by induced current flowing in the circuit. Faraday s Law gives the magnitude of induced e.m.f. while Lenz s Law gives the direction of the induced e.m.f.. BP3 3 FYSL
Example The solenoid in figure is moved at constant velocity towards a fixed bar magnet. Using Lenz s law, determine the direction of the induced current through the resistor. Solution Figure shows a permanent magnet approaching a loop of wire. The external circuit attached to the loop consists of the resistance R. Find the direction of the induced current and the polarity of the induced e.m.f.. Exercise A bar magnet is held above a loop of wire in a horizontal plane, as shown in figure. The south end of the magnet is toward the loop of the wire. The magnet is dropped toward the loop. Find the direction of the current through the resistor a. while the magnet falling toward the loop and b. after the magnet has passed through the loop and moves away from it. BP3 4 FYSL
d 5..3 Use induced emf 5..4 Derive and use induced e.m.f. in straight conductor/ in coil/ in rotating coil d According to Faraday s law, the magnitude of induced e.m.f. is given by CASE 1: Induced e.m.f. in a straight conductor Consider a linear (straight) conductor PQ of length l is moved perpendicular with velocity v across a uniform magnetic field B. When the conductor moved through a distance x in time t, the area swept out by the conductor is given by BAcos Blx d dblx dx Bl Blv and θ = 0 and In general, the magnitude of the induced e.m.f. in a linear conductor is given by vbl sin where θ is the angle between v and B In vector form, d v Bd l ε The direction of induced e.m.f. can be determined by using right hand rule. Thumb induced e.m.f./ induced current Other fingers direction of motion Palm magnetic flux v B BP3 5 FYSL
CASE : Induced e.m.f. in coil Figure shows a coil of N turns each of area A in a magnetic flux density B. Since, an e.m.f. can be induced in three ways: 1. by changing the magnetic flux density B. by changing the area A of the loop in the field 3. by changing the orientation θ with respect to the field (rotating coil) By changing the magnetic flux density B For a coil of N turns: By changing the area A For a coil of N turns: d N and BAcos d N and BAcos d BAcos N By changing the magnetic flux density B d BAcos N By changing the area A of the loop in uniform magnetic field NA cos db If B is perpendicular to the plane of coil θ = 0 NA db NB cos da If B is perpendicular to the plane of coil θ = 0 NB da For a coil is connected in series to a resistor of resistance R and the induced emf exist in the coil as shown in figure. d N and IR d IR N BP3 6 FYSL
CASE 3: Induced e.m.f. in a rotating coil As a coil rotates in a uniform magnetic field, the magnetic flux through the area enclosed by the coil changes with time; therefore induces an e.m.f. & current in the coil according to Faraday s Law. From: Φ = NBA cos θ As coil rotates, θ change, flux changes cos θ, Φ Flux changes induces an emf or current Suppose that, coil has N turns, all of the same area A & rotates in a magnetic field B with a constant angular velocity ω: d N d BAcos N BAcos d NAB cos and θ = ωt in rotational motion NAB d and cost NAB sint The magnitude of the e.m.f induced in rotating coil is given by: NAB sint or NAB sin Coil perpendicular with B θ = 0 ε min = 0 Coil parallel with B θ = 90 ε max = NABω BP3 7 FYSL
Example A 0 cm long metal rod CD is moved at speed of 5 m s -1 across a uniform magnetic field of flux density 50 mt. The motion of the rod is perpendicular to the magnetic field as shown in figure below. Solution a) Calculate the motional induced e.m.f. in the rod. b) If the rod is connected in series to the resistor of resistance 15 Ω, determine i. the induced current and its direction. ii. the total charge passing through the resistor in two minute. A single turn circular shaped coil has resistance of 10 Ω and area of its plane is 5.0 cm. It moves towards the north pole of a bar magnet as shown in figure below. If the average rate of change of magnetic flux density through the plane of the coil is 0.50 T s -1, determine the induced current in the coil and state the direction of the induced current observed by the observer shown in figure above. A narrow coil of 10 turns and diameter of 4.0 cm is placed perpendicular to a uniform magnetic field of 1.0 T. After 0.5 s, the diameter of the coil is increased to 5.3 cm. a. Calculate the change in the area of the coil. b. If the coil has a resistance of.4 Ω, determine the induced current in the coil. BP3 8 FYSL
Example A rectangular coil of 00 turns has size 10 cm x 15 cm. It rotates at a constant angular velocity of 600 r.p.m. in a uniform magnetic field of flux density 0 mt. Calculate a. the maximum e.m.f. produced by the coil. a. the induced e.m.f. at the instant when the plane of the coil makes an angle of 60 with the magnetic field. Solution Exercise A linear conductor of length 0 cm moves in a uniform magnetic field of flux density 0 mt at a constant speed of 10 m s -1. The velocity makes an angle 30 to the field but the conductor is perpendicular to the field. Determine the induced e.m.f. across the two ends of the conductor. Answer:.0 10 - V A flat coil having an area of 8.0 cm and 50 turns lies perpendicular to a magnetic field of 0.0 T. If the flux density is steadily reduced to zero, taking 0.50 s, find a. the initial flux through the coil. b. the initial flux linkage. c. the induced e.m.f. Answer: 1.6 10-4 Wb; 8.0 10-3 Wb; 1.6 10 - V A circular shaped coil 3.0 cm in radius, containing 0 turns and have a resistance of 5.0 W is placed perpendicular to a magnetic field of flux density of 5.0 x 10-3 T. If the magnetic flux density is reduced steadily to zero in time of.0 ms, calculate the induced current flows in the coil. Answer:.83 10 - A The flexible loop has a radius of 1 cm and is in a magnetic field of strength 0.15 T. The loop is then stretched until its area is nearly zero. If it takes 0.0 s to close the loop, find the magnitude of the average induced e.m.f. in it during this time. Answer: 3.4 10 - V A circular coil has 50 turns and diameter 1.0 cm. It rotates at a constant angular velocity of 5 rev s -1 in a uniform magnetic field of flux density 50 mt. Determine the induced e.m.f. when the plane of the coil makes an angle 55 to the magnetic field. Answer : 1.77 x 10-5 V A coil of area 0.100 m is rotating at 60.0 rev s -1 with the axis of rotation perpendicular to a 0.00 T magnetic field. If the coil has 1000 turns, find the maximum e.m.f. generated in it. Answer: 7.54 kv BP3 9 FYSL
5.3 Self-inductance Self-induction is defined as the process of producing an induced e.m.f. in the coil due to a change of current flowing through the same coil. I I When the switch is closed, a current begin to flow in the solenoid. The current produces a magnetic field lines through the solenoid and generate the magnetic flux linkage. If the resistance of the variable resistor changes, thus the current flows in the solenoid also changed, then so does the magnetic flus linkage. According to Faraday s law, an e.m.f has to be induced in the solenoid itself since the flux linkage changes. In accordance to the Lenz s law, the induced e.m.f opposes the change that has induced it and it is therefore known as back e.m.f. I increases I decreases S N N S ε induced ε induced N S N S If the current is increasing, so is the magnetic flux. According to the Lenz s law, the induced e.m.f. acts to oppose the increasing flux, which means it acts like a source of e.m.f. that opposes the external e.m.f.. This induced e.m.f. is also known as back e.m.f.. Therefore the direction of the induced e.m.f is in the opposite direction of the current I. If the current is decreasing, so is the magnetic flux. According to the Lemz s law, the induced e.m.f. acts to oppose the decreaseing flux, which means it acts to bolster the flux, like a source of e.m.f. reinforcing the external e.m.f.. Therefore the direction of the induced e.m.f is in the same direction of the current I. BP3 10 FYSL
5.3.1 Define self-inductance 5.3. Apply self-inductance equation for coil and solenoid From the process of self-induction, we know that magnetic field B is proportional to current I, and magnetic flux Ф is proportional to magnetic field B. Therefore Mathematically, I LI Self-inductance L is defined as the ratio set induced e.m.f. to the rate change current in the coil. L It is a scalar quantity and its unit is henry (H). di Unit conversion: 1 H 1 Wb A 1 1 T m A 1 For N turns of coil: Magnetic flux linkage N LI L N I 0 N r A For N turns of solenoid: L N I 0 N l A The value of the self-inductance depends on the size and shape of the coil, the number of turn (N), the permeability of the medium in the coil (). A circuit element which possesses mainly self-inductance is known as an inductor. It is used to store energy in the form of magnetic field. The symbol of inductor: BP3 11 FYSL
Example At an instant, the current in an inductor increases at the rate of 0.06 A s -1 and back e.m.f. of 0.018 V was produced in the inductor. a. Calculate the self-inductance of the inductor. b. If the inductor is a solenoid with 300 turns, find the magnetic flux through each turn when the current of 0.80 A flows in it. Solution A 500 turns of solenoid is 8.0 cm long. When the current in the solenoid is increased from 0 to.5 A in 0.35 s, the magnitude of the induced e.m.f. is 0.01 V. Calculate a. the inductance of the solenoid, b. the cross-sectional area of the solenoid, c. the final magnetic flux linkage through the solenoid. (Given µ 0 = 4p 10-7 H m -1 ) Exercise The coil in an electromagnet has an inductance of 1.7 mh and carries a constant direct current of 5.6 A. A switch is suddenly opened, allowing the current to drops to zero over a small interval of time, t. If the magnitude of the e.m.f. induced during this time is 7.3 V, what is t? Answer: 1.3 ms A 500 turns solenoid is 8.0 cm long. When the current in this solenoid is increased from 0 to 0.5 A in 0.35 s the magnitude of the induced e.m.f. is 0.01 V. Find a. the inductance and b. the cross sectional area of the solenoid. Answer: 1.7 mh; 4.3 10-4 m A 40.0 ma current is carried by a uniformly wound air-core solenoid with 450 turns, a 15.0 mm diameter and 1.0 cm length. Calculate a. the magnetic field inside the solenoid, b. the magnetic flux through each turn, c. the inductance of the solenoid. Answer: 1.88 10-4 T; 3.33 10-8 Wb; 3.75 10-4 H BP3 1 FYSL
5.4 Energy stored in inductor 5.4.1 Derive and use the energy stored in an inductor Consider a coil of self-inductance L. Suppose that at time t the current in the coil is in the process of building up to its stable value I at a rate di/. The magnitude of the back e.m.f. ε is given by di L The power P in overcoming this back e.m.f. is given by P I Power time = Energy di P IL P ILdI du ILdI du L I 0 IdI 1 U LI Example An 8.0 cm long solenoid with an air-core consists of 100 turns of diameter 1. cm. If the current flows in it is 0.77 A, determine a. the self-inductance of the coil b. the energy stored in the coil (Given µ 0 = 4π x 10-7 H m -1 ) Solution Exercise At the instant when the current in an inductor is increasing at a rate of 0.064 A s -1, the magnitude of the back e.m.f. is 0.016V. a) Calculate the inductance of the inductor. b) If the inductor is a solenoid with 400 turns and the current flows in is 0.70 A, determine i. the magnetic flux through each turn ii. the energy stored in the solenoid. Answer: 0.5 H; 4.5 10-4 Wb, 6.48 10 - J BP3 13 FYSL
5.5 Mutual-inductance Mutual induction is defined as the process of producing an induced e.m.f in one coil due to the change of current in another coil. Primary coil Secondary coil Consider two circular close-packed coils near each other and sharing a common central axis as shown in figure. A current I 1 flows in coil 1, produced by the battery in the external circuit. The current I 1 produces a magnetic field lines inside it and this field lines also pass through coil as shown in figure. If the current I 1 changes with time, the magnetic flux through coils 1 and will change with time simultaneously. Due to the change of magnetic flux through coil, an e.m.f. is induced in coil. This is in accordance to the Faraday s law of induction. In other words, a change of current in one coil leads to the production of an induced e.m.f. in a second coil which is magnetically linked to the first coil. According to Lenz s law, the induced current produced in coil will oppose the change in I 1. This process is known as mutual induction. At the same time, the self-induction occurs in coil 1 since the magnetic flux through it changes. S N S N S N N S BP3 14 FYSL
5.5.1 Define mutual inductance 5.5. Use mutual inductance question between two coaxial solenoids or a coaxial coil and solenoid If the current I 1 in coil 1 is changes, the magnetic flux B through coil will change with time t and an induced e.m.f ε will occur in coil where Mathematically, Lenz s law di 1 M 1 di1 If vice versa, the induced e.m.f. in coil 1, ε 1 is given by Conclusion, 1 M 1 di M 1 M 1 M Mutual inductance is defined as the ratio of induced e.m.f in a coil to the rate of change of current in another coil. For a given pair of coils, the value of mutual inductance is the same and does not depend on which coil carries the current and which coil experiences induction. For N turns of coil: M 1 N I or 1 M 1 N1 I 1 Mutual inductance between two coaxial solenoids or a coaxial coil and solenoid M 0 N1N A l N 1 : primary coil N : secondary coil BP3 15 FYSL
Example A current of.0 A flows in coil P and produced a magnetic flux of 0.6 Wb in it. When a coil S is moved near to coil P coaxially, a flux of 0. Wb is produced in coil S. Given that, coil P has 100 turns and coil S has 00 turns. a. Calculate self-inductance of coil P and the energy stored in P before S is moved near to it. b. Calculate the mutual inductance of the coils. c. If the current in P decreasing uniformly from.0 A to zero in 0.4 s, calculate the induced e.m.f. in coil S. Solution Primary coil of a cylindrical former with the length of 50 cm and diameter 3 cm has 1000 turns. If the secondary coil has 50 turns, calculate : a. its mutual inductance b. the induced e.m.f. in the secondary coil if the current flowing in the primary coil is changing at the rate of 4.8 A s -1. Exercise Two coils, X and Y are magnetically coupled. The e.m.f. induced in coil Y is.5 V when the current flowing through coil X changes at the rate of 5 A s -1. Determine: a. the mutual inductance of the coils b. the e.m.f. induced in coil X if there is a current flowing through coil Y which changes at the rate of 1.5 A s -1. Answer : 0.5 H ; 0.75 V Two coils, X and Y have mutual inductance of 550 mh. Determine the rate of change of magnetic flux through coil Y at the instant when the current flowing through coil X changes at the rate of 5.5 A s -1. Given that, both coil X and Y has 100 turns. Answer: 3 10 - Wb s -1 BP3 16 FYSL