 phase
difference between current and voltage. ε = εmSinωt, I= ImSin(ωt+Φ) 23 Average power loss over a complete cycle in an LCR circuit is P = εrmsIrmsCosΦ * In a purely resistive circuit Φ = 0; P = VRMSIRMS. * In a purely inductive circuit Φ = Π/2; P = 0. * In a purely capacitive circuit Φ = Π/2; P = 0. 24 In an LCR circuit, the circuit admits maximum current if XC = XL, so that Z = 1 1 R and resonant frequency 𝜔𝑟 = 𝑎𝑛𝑑𝜗𝑅 = √𝐿𝐶
2𝜋√𝐿𝐶
25 Q factor of series resonant circuit is defined as the ratio of voltage developed across the inductance or capacitance at resonance to the applied voltage across ‘R’, 𝜔 𝐿 1 𝜔 Q= 𝑟 𝑜𝑟 also 𝑄 = 𝑟 where 2∆𝜔 is bandwidth. 𝑅
26
𝜔𝑟 𝐶𝑅
for a transformer,
2∆𝜔
Es N s i p K E p N p is
In an ideal transformer, εPIP = εSIS. i.e If NS>NP; εS>εP& IS
NS; εP>εS & IP
1 q0 dt 2 LC
CONCEPT MAP
EMI and application
59
60
61
QUESTIONS MAGNETIC FLUX, INDUCED E.M.F, 1
Two concentric circular coils are perpendicular to each other. Coil I 1 carries a current i. If this current is changed, will this induce a current in the coil II?
I
II
2
3
[No- Field due to one coil is parallel to the plane of the second coil. So flux does not change.] A closed loop of wire is being moved with constant velocity without 1 changing its orientation inside a uniform magnetic field. Will this induce a current in the loop? [Ans: No there is no change in ΦB] A cylindrical bar magnet is kept along the axis of a circular coil and near it 1 as shown in the fig. Will there be any induced current at the terminals of the coil when the magnet is rotated a) about its own axis b) about an axis perpendicular to the length of the magnet?
N Fig (i)
S Fig(ii)
Ans Fig. (i) No e.m.f will be induced, as these is no change in flux. 62
Fig (ii) Yes, Φ changes continuously. So e.m.f is induced in the coil. 4
A conducting wire is kept along the N→S direction and is allowed to fall freely. Will an e.m.f be induced in the wire? (Yes) 5 A conducting wire is kept along the E→W direction and is allowed to fall freely. Will an e.m.f be induced in the wire? (Yes) 6 A vertical magnetic pole falls down through the plane of magnetic meridian. Will any e.m.f be induced between its ends? Ans: No, because the pole intercepts neither Bv or BH 7 A wheel with a certain number of spokes is rotated in a plane normal to earth’s magnetic field so that an emf is induced between the axle and rim of the wheel, keeping all other things same, number of spokes is changed. How is the e.m.f affected? (Hint: Number of spokes does not affect the net emf) 8 What are eddy currents? 9 Explain any two applications of eddy current. 10 The magnetic flux linked with a coil passing perpendicular to the plane of the coil changes with time Φ = 4t2 + 2t + 3, where “t” is the time in seconds. What is magnitude of e.m.f induced at t = 1 second? Ans: (e = dΦ/dt =
d 4t 2 2t 3,e = 8t +2 dt
If t = 1s
1
1
1
1
1 2 3
e= 10V)
11 A wheel fitted with spokes of radius ‘r’ is rotating at a frequency of n revolutions per second in a plane perpendicular to magnetic field B Tesla. What is the e.m.f induced between the axle and rim of the wheel [2] Φ = BA
3
e = d(BA)/dt= B dA/dt, dA/dt= Πr2x n e = B. Πr2n 12 Two coils P and S are arranged as shown in the figure. (i) What will be the direction of induced current in S when the switch is closed? (ii) What will be the direction of induced current in S when the switch is opened?
63
2
P
S
Ans: (i) anticlockwise (ii) clockwise 13 A conducting circular loop is placed in a uniform magnetic field B = 2 0.020T with its plane perpendicular to the field. Somehow, the radius of the loop starts shrinking at a constant rate of 1mm/s. Find the induced current in the loop at an instant when the radius is 2cm. Ans. (Ф= Πr2B d Ф/dt = 2ΠrB dr/dt e= 25μV 14 A 12V battery is connected to a 6Ω; 10 H coil through a switch drives a constant current in the circuit. The switch is suddenly opened. Assuming that it took 1ms to open the switch calculate the average e.m.f induced across the coil. Ans. (I initial=2A I final= 0 𝜀=-Ldi/dt = 20000V) 15 A coil of mean area 500 cm2 having 1000 turns is held perpendicular to a uniform magnetic field of 0.4 G. The coil is turned through 180 o in 1/10 seconds. Calculate the average induced e.m.f. Ans. (0.04 V) 16 A conducting rod of length l with one end pivoted is rotated with a uniform angular speed ω in a Vertical plane normal to uniform magnetic field B. Deduce an expression for e.m.f induced in this rod. 17 Two identical co-axial coils carry equal currents. What will happen to the current in each loop if the loops approach each other? (2)
2
2
2
2
Ans. (Acc to Lenz’s law current in each coil will decrease) 18 Obtain the direction of induced current and e.m.f when the conductor AB is moved at right angles to a stationary magnetic field (i) in the upward direction (ii) in the downward direction. (i) B to A (ii) A to B) 64
2
B
N
S
G
A
19 A fan blade of length 0.5 m rotates perpendicular to a magnetic field of 5x10 -5 T. If the e.m.f induced between the centre and the end of the blade is 10 -2 V.Find the rate of rotation. Ans. (e=B dA/dt ; dt= 1/n ; n=254.7 rev/s)
3
20 The figure shows a square loop having 100 turns an area of 2.5x10 -3 m2 3 and a resistance of 100Ώ. The magnetic field has a magnitude of B= 0.4 T. Find the work done in pulling the loop out of the field slowly and uniformly in 1 second. P
Q
R
* * * * * * * * * * * * *v * * *
* * * *
Also draw graph showing the variation of power delivered when the loop is moved from P to Q to R.
(1x 10-6J)
21 Two coils have a mutual inductance of 0.005H. The current changes in the 3 first coil according to the equation I= I0 Sin ωt where I0 =10A and ω=100∏ rad/s. Calculate the maximum value of e.m.f in the second coil. (5 π volts) 22 A long rectangular conducting loop of width L mass m and resistance R is 3 placed partly above and partly below the dotted line with the lower edge parallel to it. With what velocity it should continue to fall without any acceleration?
65
* * * * * * * * *
* * * ** * * * * -*--*--*----------*---*---(mg = B2l2v/r ; v=mgr/ B2l2 ) INDUCTANCE 1
Two conducting circular loops of radii R1 and R2 are placed in the same plane with their centres coinciding. Find the mutual inductance between them assuming R2<< R1. (M=µ0 πR22 /2R1)
2
2
Prove that the total inductance of two coils connected in parallel is
2
1 1 1 LT L1 L2
3
4
5
Two circular loops are placed with their centres at fixed distance apart. 2 How would you orient the loops to have (i) maximum (ii) minimum Mutual inductance? A coil of wire of certain radius has 600 turns and inductance of 108mH.