Moving Charges and Magnetism


   
 
Evaluation of Magnetic Field
Magnetic Field due to a Straight Conductor Carrying Current
 
Consider a straight conductor XY carrying a current I as shown. To find the magnetic field at P, consider a small current element of length dl.
 
If be the position of point P from current element q the angle between
 
Let CO = I
 
According to Biot and Savart's law, field at P due to current flowing straight through the conductor XY is
 
 
 
 
 
 
Substituting we get,
 
 
 
 
 
Note:
 
For an infinitely long conductor,
 
 
 
 
 
Conclusion
 
The magnetic field B is
 
a) Proportional to the current I.
 
b) Inversely proportional to the distance d
 
c) The field has the same value on a circle of radius 'a'. Hence the magnetic field around the wire is in form of concentric circles.
 
d) When a 0, the value of magnetic field strength becomes large.
 
e) Is in a direction perpendicular to both the straight wire and the radius vector.
 
The direction of B can be given by the right hand thumb rule as shown in the diagram below:
 
 
Magnetic Field due to a Circular Coil Carrying Current
Consider a circular coil of radius r carrying current I in direction shown. If the coil lies in the plane of paper, the field at the center of the coil is obtained as follows:
 
 
Here q = 90o as seen therefore sin q = 1
 
 
 
Total magnetic field at the centre due to the circular coil of length 2pr would be
 
 
On integrating,
 
 
Note:
 
If there were N turns, then
 
 
Magnetic Field on the Axis of a Circular Current Loop
Consider a circular coil of radius 'a' with centre O. Let the plane of the coil be perpendicular to the plane of paper and a current I flowing in the coil. P is the point at a distance x from 'O'. OP = x.
 
Consider two small elements of the coil each of length dl, at a A and B which are situated diametrically opposite.
 
 
 
According to Biot Savarts' Law, magnetic field due current element Idl at A is
 
 
 
The direction of is perpendicular to    and so acts along PC which is perpendicular to PA.
 
Similarly the magnetic field due to current element Idl at B is
 
 
The direction of dB' is along PD.
 
On resolving dB and dB' into two perpendicular components only sinf components contribute to the magnetic field at 'P'. The cosf components cancel each other.
 
The total magnetic field induction due to the current through the whole circular coil is
 
 
 
 
 
 
 
 
for N turns
 
 
Note:
 
1) When x = 0, P lies at the centre of the coil. So,
 
 
2) When x >> a
 
 
 
 
Where
 
pm = N I A is called the magnetic dipole moment of the current loop.
 
 
Therefore, a current loop can be regarded as a magnetic dipole which produces its magnetic field.
 
S.I unit of magnetic dipole moment is A - m2.
 
Magnetic dipole moment pm is the product of current and area of current loop.
 
Such current loop behaving like magnets have N and S polarity. If the direction of current through the coil is clockwise, then that face of loop is south pole. If the current direction is anticlockwise then that face of loop is north pole.
 
 
D (Symbols) depicts the direction of conventional currents that is anticlockwise in circular coil. Since it generates a magnetic field around it, the upper face of the coil as in (a) is the North Pole. If the current is in clockwise direction, then the upper face of the coil (b) is the South Pole.
 
  
     
   
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