Example.1. Two infinite conducting planes at z=0 and z=d carry currents in opposite directions with surface current density in opposite directions Calculate the magnetic field everywhere in space.
 Let us take the yz plane to be the plane of the paper so that the direction of current flow is perpendicular to this plane. Consider a single conducting plane carrying current in the x direction. Let the plate be in the xy plane. The surface current is shown by a circles with a dot.
 Consider an Amperian loop of length L and width 2z located as shown. The magnetic field direction above the plane is in –y direction while that below is in +y direction. Taking the loop to be anticlockwise, the line integral of magnetic field is
 Thus, above the plane the field is and below the plane it is Not that the field strength is independent of the distance from the plane.
 If we have two planes, at z=0 and z=d, the former having a linear current density the fields would add in the region between the plates and would cancel outside. Between the plates, and outside both the plates the field is zero.
Example.2. Part of a long current carrying wire is bent in the form of a semicircle of radius R. Calculate the magnetic field at the centre of the semi circle.
 Since the centre is along the line carrying the current, for the straight line section and the contribution to magnetic field is only due to the semicircular arc.
 The field is into the page at B and is given by Biot Savart’s law to be
Example.3. A long cylindrical wire has a current density flowing in the direction of its length whose density is J = J_{0}r, where r is the distance from the cylinder’s axis. Find the magnetic field both inside and outside the cylinder.
 The total current in the wire can be obtained by integrating the current density over its cross section The current enclosed within a radius r from the axis is
 Using Ampere’s law, the field at a distance is r < R is
 The total current carried by the wire being the field outside is given by
Example.4. A thin plastic disk of radius R has a uniform charge density σ. The disk is rotating about its axis with an angular speed ω. Find the magnetic field along the axis of the disk at a distance z from the centre.
 The current on the disk can be calculated by assuming the rotating disc to be equivalent to a collection of concentric current loops.
 Consider a ring of radius r and width dr. As the disc rotates, the rotating charge on this annular section behaves like a current loop carrying current The field at a distance z due to this ring is
 The total field is obtained by integrating this expression from 0 to R,
which can be easily performed by a substitution x = r^{2} + z^{2}. The result is
Example.5. Two infinite conducting planes at z=0 and z=d carry currents in opposite directions with surface current density in the same directions Calculate the magnetic field everywhere in space.
 The field due to a single plane carrying a linear current density direction above the plane while below the plane it is in +y direction.
 Since the magnitude of the field is independent of distance, the field cancels between the planes and add up above the planes. For z > d, we have while for z<0 it is
 Its value between the planes is zero.
Example.6. Two infinite conducting sheets lying in xz and yz planes intersect at right angles along the z axis. On each plane a surface current Find the magnetic field in each of the four quadrants into which the space is divided by the planes.
 The field due to the current in xz plane is along the plane (i.e. for y > 0) and along the plane (i.e. for y < 0)
 Likewise for the yz plane, the field is along the plane (i.e. for x > 0) and along below (x < 0). The magnitude of the field in each case is the same and is
 Thus the field along the first quadrant is and along the fourth is One can similarly find the fields in the other two quadrants.
Example.7. Consider the loop formed by two semicircular wires of radii R_{1} and R_{2} (>R_{1}) and two short straight sections, as shown. A current I flows through the wire. Find the field at the common centre of the semicircles.
 From example 2 it follows that the contribution to the field due to two straight sections is zero.
 The smaller semicircle section gives a field into the page while the bigger semicircle gives a field out of the page. The net field is (into the page)
Example.8. The current density along a long cylindrical wire of radius a is given by where r is the distance from the axis of the cylinder. Use Ampere’s law to find the magnetic field both inside and outside the cylinder.
 First find the current enclosed within a distance r from the axis,
 Thus the field inside is given by
 Outside the cylinder the total current
11 videos45 docs73 tests

1. What is magnetostatics? 
2. What is the difference between current and current density? 
3. What is BiotSavart's Law and Ampere's Law? 
4. How is the magnetic field calculated for a long straight wire carrying current? 
5. What is the magnetic field along the axis of a circular loop carrying current? 
11 videos45 docs73 tests


Explore Courses for Electrical Engineering (EE) exam
