Magnetism & Moving Charge
OERSTED'S EXPERIMENT
The first evidence of the relationship of magnetism to moving charges was discovered in 1819 by the Danish scientist Hans Christian Oersted. He found that a compass needle was deflected by a current carrying wire. When the direction of current was reversed the deflection also got reversed.
BIOT-SAVART'S LAW
It turns out that there is no source of magnetic force similar in nature to point electric charges giving rise to electric fields. Such magnetic monopoles have not been found so far. The elementary source of magnetic force is a current element
1. The magnetic field grows weaker as we move farther from its source. In particular, the magnitude of the magnetic field dB is inversely proportional to the square of the distance from the current element
2. The larger the electric current, the larger is the magnetic field. In particular, the magnitude of the magnetic field
current I.
3. The magnitude of the magnetic field
4. The direction of the
These features of field
Here,
The constant
The magnetic field
Its SI unit is tesla (T) which is equivalent to Wb/m2.
The dimensions of
The magnitude of
1. Field due to Straight Current-Carrying Conductor
According to Biot-Savart law,
⸫
Case (A) : The point P is along the length of the wire, as in Fig. (A). Then
Case (B) : The point P is at a distance d from the wire, as shown in Fig. (B). Every current element
But
⸫
or
Note that
(1) For points along the length of the wire (but not on it) the field is always zero.
(2) For points at a perpendicular distance d from the wire, field B varies inversely with distance,
(3) The field is always perpendicular to the plane containing the wire and the point. So in a plane perpendicular to the wire the lines of force are concentric circles.
(4) If the wire is of infinite length and the point P is not near its any end,
(5) If the point is near one end of an infinitely long wire,
(6) If the wire is of finite length and the point is on its perpendicular bisector,
(7) If the wire is of finite length and the point is near its one end,
Application 1
Figure shows two long straight wires carrying electric currents of 10 A each, in opposite directions. The separation between the wires is 5.0 cm. Find the magnetic field at a point P midway between the wires.
Solution:
The right-hand thumb rule shows that the magnetic field at P due to each of the wires is perpendicular to the plane of the diagram and is going into it. The magnitude of the field due to each wire is
Total field due to both the wires is
Application 2
A pair of stationary and infinitely long bent wires are placed in the x-y plane as shown. The wires carry currents of 10 amperes each as shown. The segments L and M are along the x-axis. The segments P and Q are parallel to the y-axis such that OS = OR = 0.02 m. Find the magnitude and direction of the magnetic induction at the origin O.
Solution:
As point O is along the length of segments L and M so the field at O due to these segments will be zero. Further, as the point O is near one end of a long wire,
so
Substituting the given data,
2. Field due to a Circular Current-Carrying Segment at its Centre
Let AB be a circular segment of radius R. Point P is at its centre. Here,
(i) each element is at the same distance from the centre, i.e., r = R = constant,
(ii) the angle between element
(iii) the contribution of each element to
⸫
But
⸫
Note that
(1) The angle
(2) If the loop is semicircular (i.e,
(3) If the loop is a full circle with N turns (i.e.,
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