MHT-CET : Physics Entrance Exam

### MHT - CET : Physics - Electomagnetic Induction Page 7

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Resonant Circuits:

• The current i in this circuit is given by

i = i0 sin wt. ……. (1)

• The e.m.f. e at any instant is the sum of voltage drops across R, across L and across C.
 \ e = L di + Ri + q ...(2) dt C
•
• Substituting (1) in (2) we get,

\ e = I0 [ R sin wt + (XL - Xc ) cos wt ] … (3)
• Multiplying and dividing R.H.S. of (3) by
e = I0 x

 · Let tan f XL - Xc , so that sin f = XL - Xc R
•
 and cos f = R

·  substituting we get e - E0 sin [ wt + f ]
where E0 = I0

Impedance Z:

 z = e0 = I0 =

Phase Difference : The instantaneous current and e.m.f. are
given by i = I0 sin wt and e = e0 sin (
wt + f ).

These equations show that e.m.f. e leads the current by a phase angle
f given by

 f = tan -1 XL - XC R

Resonant Frequency:

• When the impedance offered by the circuit is minimum, the current through the circuit is maximum. This condition is known as series resonance.
• Z = is minimum when XL = XC and
the minimum value of Z = R.

Series resonance curve

Þ

XL = XC

 wL = 1 wC

 Þ w2= 1 LC

 Þ 4p2f2r = 1 LC

 Þ fr2 = 1 4pLC

 Þ fr = 1 2p

...(Resonance frequency)

L-C-R Parallel Resonance:

L - Inductance,
C - Capacitance,
e - Applied alternating e.m.f.

• Current in inductor lags behind e.m.f. by 90°
 \ iL = e0 sin (wt- p / 2) XL
•
• Current in capacitor leads e.m.f. by 90°
 \ iC = e0 sin (wt+ p / 2) XC
•
• Current drawn from source
 i = iL + iC = e0 ( 1 - 1 ) cos wt. XC XL
•
• If XL = XC, i = 0 No current is drawn from the source, alternating current goes on circulating in the LC loop. This condition is called the parallel resonance.
• Parallel resonance curve

At resonance,

XL= XC

 wL = 1 wC

 \ w = 1

 Resonant fresquency fr = w = 1 2p 2p

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