Current Carrying Conductors
Consider C1 D1 and C2 D2 two infinite long straight conductors carrying currents I1 ad I2 in the same directing they are held parallel to each other at a distance r apart in the plane of paper. The magnetic field is produce due to current through each conductor shown separately in and (b). Since each conductor is in the magnetic field produced by the other therefore each conductor experiences a force.
Magnetic field induction at a point p one conductor C2 D2 due to current I1 passing through C1 D1 is given by B1 = μ0 / 4 π 2 I1 / r (1)
According to right hand rule the direction of magnetic field B1 is perpendicular to the plane of paper directed inwards.
As the current carrying conductor C2 D2 lies in the magnetic field B1 (produced by the current through C1D1 therefore, the unit length of C2 D2 will experience a force given by
F2 = B1 I2 X 1 = B1 I2
Putting the value of B1 we have
F2 = μ0 / 4 π 2 I 1 I2 / r (2)
According to Fleming's left hand rule force on conductor C2 D2 acts in the place of the paper perpendicular to C2 D2 directed towards C1 D1 similarly it cane shown that the conductor C1 D1 also experiences a force given by eqn. (2) which acts into he lance of paper perpendicular to C1 D1 and directed towards C2 D2hence C1 D1 and C2 D2 attract each other.
It means the two linear parallel conductors carrying currents in the same direction attract each other. The force experienced by the unit length of each conductor is given by (2) further it can be shown that if the current in conductor C1D1 and C1 D2 are in opposite directions they repel each other with the same force as given by eq. (2)
Definition of ampere
Relation is used to define 1 ampere
Let I1 = I2 = IA; r = 1 m
Then F = μ0 / 4π 2I1 I2 / r = 10-7 x 2 x 1 x 1 / 1
(∴ μ0 / 4 π = 10-7) = 2 x 10 -7 Nm - 1
Thus one ampere is that much current which when flowing through each of the two parallel uniform long linear conductor placed in free space at a distance of one meter from each other will attract or repel each other with a force of 2 x 10-7 N per meter of the length.
Note if two liner current carrying conductors of unequal length are held parallel to each other then the force on long conductor is due short conductor and long conductors.
if I L = length of short and long conductor respectively.
I1 I2 = current through short hang long conductor
R = separation between these two parallel conducts (i) then force on long conduct = force on short conductor = μ0 / 4 π2I1 I2 / r.
(ii) The force on each conductor is same in magnitude and opposite in direction the force on two conductors obey Newton's third law of motion.
(b) When two currents approach a point or they go away from that point then they experience an attractive force.
(c) When one current out of the two approaches a point and another one goes away from that point they then experience a force of repulsion.
(d) When two charges of values q1 and q2 moving with velocities v1 and v2 on parallel straight paths distance r apart then the force acting between them is given by
F = μ0 / 4 π q1 q2 v1 v2 / r2
Proof: When a charge q moving with velocity v for time dt covers a distance dl it will behave as a current element I dl where
Idl = q / dt dl = q dl / dt = qv
If I1 I2 are the currents corresponding to motion of charges q1 and q2 with velocity v1 and v2 then
I1 dl1 = q1 v and I2 dl 2 = q2 v2
The magnitude of magnetic force acting between these two parallel current elects distance r apart is given by
Fm = μ0 / 4π I1 I2 / r2 dl1 dl2 or Fm = μ0 q1 q2 / 4π r2 v1 v2.
(e) A fixed horizontal wire A carrying current I1 will support another horizontal wire B, carrying current I2hanging in air at a distance r below the wire A if the direction of currents in the two wires is same here weight pr unit length of wire B is equal to the force of attraction acting per unit length of the wire B
Mg μ0 / 4π 2I1 I2 / r
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