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A long horizontal wire AB, which is free to move in a vertical plane and carries a steady current of 20A, is in equilibrium at a height of 0.01m over another parallel wire CD which is fixed in a horizontal plane. Show that AB is slightly depressed , it executes simle harmonic motion. Find the period of oscillations.
Weight per unit length of wire is balanced by magnetic force per unit length. or, { 4¶*10-7*i1*i2 }/d2 = mg/l where i1 = Current through wire CD i2 = Current through wire AB = 20A d= Vertical separation between wires = 0.01 mtrs. Let wire AB is depressed by 'x' then, Vertical separation between wires = d-x Now magnetic force per unit length on AB = { 4¶*10-7*i1*i2 }/(d-x)2 = K/(d-x)2 where K is a constant equal to 4¶*10-7*i1*i2. or, Net vertical force per unit length on AB =[ { 4¶*10-7*i1*i2 }/(d-x)2 ] - (mg/l) = K [1/ (d-x)2 - 1/d2 ] = K [( 2dx-x2 )/{ (d-x)2 *d2 } ] Taking x/d=µ & x<<d;i.e. µ<<1 Simplifying above expression give R.H.S. = 2dx/d4 [(1-µ/2)/{(1-µ)2}]K = 2x/d3 K : Ignoring the negligible terms 1-µ & 1-µ/2 equal to 1. L.H.S = (m/l)w2x = 2x/d*( K/d2 ) ; or, w2 = 2g/d where w=angular frequency of SHM with F=constant*x(Hookes' Law) K/d2 = mg/l So,period = 2¶/w= ( 2¶ ) / [ (2g/d)1/2 ] = 0.14 sec.
Weight per unit length of wire is balanced by magnetic force per unit length.
or, { 4¶*10-7*i1*i2 }/d2 = mg/l where i1 = Current through wire CD
i2 = Current through wire AB = 20A
d= Vertical separation between wires = 0.01 mtrs.
Let wire AB is depressed by 'x' then,
Vertical separation between wires = d-x
Now magnetic force per unit length on AB = { 4¶*10-7*i1*i2 }/(d-x)2 = K/(d-x)2 where K is a constant equal to 4¶*10-7*i1*i2.
or, Net vertical force per unit length on AB =[ { 4¶*10-7*i1*i2 }/(d-x)2 ] - (mg/l) = K [1/ (d-x)2 - 1/d2 ]
= K [( 2dx-x2 )/{ (d-x)2 *d2 } ]
Taking x/d=µ & x<<d;i.e. µ<<1
Simplifying above expression give R.H.S. = 2dx/d4 [(1-µ/2)/{(1-µ)2}]K = 2x/d3 K : Ignoring the negligible terms 1-µ & 1-µ/2 equal to 1.
L.H.S = (m/l)w2x = 2x/d*( K/d2 ) ; or, w2 = 2g/d where w=angular frequency of SHM with F=constant*x(Hookes' Law)
K/d2 = mg/l
So,period = 2¶/w= ( 2¶ ) / [ (2g/d)1/2 ] = 0.14 sec.
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