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Magnetostatics
Up until now, we have been discussing electrostatics, which deals with physics of the electric field created by static charges. We will now look into a different phenomenon, that of production and properties of magnetic field, whose source is steady current, i.e., of charges in motion. An essential difference between the electrostatics and magnetostatics is that electric charges can be isolated, i.e., there exist positive and negative charges which can exist by themselves. Unlike this situation, magnetic charges (which are known as magnetic monopoles) cannot exist in isolation, every north magnetic pole is always associated with a south pole, so that the net magnetic charge is always zero. We emphasize that there is no physical reason as to why this must be so. Nevertheless, in spite of best attempt to isolate them, magnetic monopoles have not been found. This, as we will see, brings some asymmetry to the physical laws with respect to electricity and magnetism.
Sources of magnetic field are steady currents. In such a field a moving charge experiences a sidewise force. Recall that an electric field exerts a force on a charge, irrespective of whether the charge is moving or static. Magnetic, field, on the other hand, exerts a force only on charges that are movinf. Under the combined action of electric and magnetic fields, a charge experiences, what is known as Lorentz force ,
where the field is known by various names, such as, “magnetic field of induction”, “magnetic flux density”, or simply, as we will be referring to it in this course as the “magnetic field”. Note that the force due to the magnetic field is expressed as a cross product of two vectors, the force is zero when the charges are moving perpendicular to the direction of the magnetic field as well.
Current and Current Density
Let us look at the definition of current. Current is a scalar quantity which is the amount of charge that crosses the boundary of a surface of a volume per unit time, the surface being oriented normal to the direction of flow. In the steady state there is no accumulation of charge inside a volume through whose surface the charges flow in. This results in the “ equation of continuity”.
Magnetostatics
Lecture 23: Electromagnetic Theory
Professor D. K. Ghosh, Physics Department, I.I.T., Bombay
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If is the charge density, the current is given by
where is the current density. Recalling that , we get,
Here, the minus sign is taken because, we define the current to be positive when it flows from outside the volume to the inside and the surface normal, as in the previous lectures, is taken to be outward. Using the divergence theorem, we can rewrite this as
Since the relationship is true for arbitrary volume, we can equate the integrands from both sides, which results in the equation of continuity :
The principle is a statement of conservation of electric charge.
As we stated earlier in the lecture, magnetostatics deals with steady currents which implies
so that for magnetostatic phenomena, we have. Though we will not be talking much about it, we would like to mention that this equation is also relativistically invariant.
Biot- Savart’s Law
Recall that the electric field at a point due to a charge distribution was calculated by the principle of superposition of fields due to charges inside infinitesimal volume elements which make this charge distribution. The field due to the infinitesimal charge element is given by Coulomb’s law, which is an inverse square law. We take a similar approach to calculate the magnetic field due to a charge distribution.
We consider a current distribution to comprise of infinitesimal current elements whose direction is
taken along the direction of the current flow. If is such a current element located at the position , the field due to it at the position is given by the law of Biot and Savart,
As divergence of a curl is zero, we have,
This is the magnetostatic Gauss’s law. Comparing with the corresponding electrostatic formula
, we see that this equation implies non-existence of magnetic monopoles.
We are aware that a vector field is uniquely given by specifying its divergence and curl. We will now try to find the curl of the magnetic field.
Using the form of given in eqn. (1) we get on taking curl,
Use the identity,
Let us examine each of the terms.
The first term can be simplified as follows. We first take the divergence operator inside the integral as the integration is with respect to the primed variable,
where we have written the gradient with respect to the primed variable by incorporating a minus sign
because it acts on the difference. Using the identity
The first term can be converted to a surface integral, using the divergence theorem and the surface can be taken to infinite distances making the integral vanish. The second term also vanishes because of continuity equation. Thus the first tem of eqn. (2) vanishes, leaving us with the second term
The operator can be taken inside the integral and as it is derivative with respect to r, it acts only on
,
This is known as Ampere’s Law. Thus the magnetic field is specified by
and.
Note that for the case of electric field, the curl is zero , which is characteristic of a conservative field.
We will now provide an integral formulation of these two relations.
Consider a closed volume V defined by a surface S. The normal to the surface is defined in the usual way. We can take the volume integral of the first relation and convert to a surface integral using the divergence theorem.
This is the integral form of magnetostatic Gauss’s law.
Take the second relation. Let S be an arbitrary surface through which the current passes. The surface integral is then given by
The first relation can be converted to a line integral using the Stoke’s thorem,
Let us take along the x-axis which is also the direction of the current. We will calculate the magnetic field at a distance r from the wire along the y-axis. Note that this is quite general as from whichever point we want to calculate the magnetic field, we can drop a perpendicular on to the x-axis and call this
to be the y axis. Thus the vector is in the xy plane. We have,
It is directed along because the vectors and are in the xy plane. We also have,
which gives and.
Thus the magnetic field is given by
Field along the axis of a circular loop
Let us take the loop in the xy plane. We wish to calculate the field along the z axis. A length element
along the circle is given by and has the position vector
O
P
z
θ (^) dθ
x
y
We take the point P along the z axis so that. With these relations, we have,
Thus, the magnetic field along the axis is given by
where we have used the magnetic moment vector , given by the product of the current with the area of the loop directed along the normal to the loop according to the right hand rule. It is observed that at large distances from the loop the magnetic field varies as the inverse cube of distance from the loop,
Tutorial Assignment
- 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.
- 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.
Magnetostatics
Lecture 23: Electromagnetic Theory
Professor D. K. Ghosh, Physics Department, I.I.T., Bombay
P
- 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
- 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
The total current carried by the wire being , the field outside is given by
- 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. The result is
P
Self Assessment Quiz
- 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.
- Two infinite conducting sheets lying in x-z and y-z planes intersect at right angles along the z axis. On each plane a surface current flows. Find the magnetic field in each of the four quadrants into which the space is divided by the planes.
- Consider the loop formed by two semicircular wires of radii and and two short straight sections, as shown. A current I flows through the wire. Find the field at the common centre of the semicircles.
- 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.
Solutions to Self Assessment Quiz
- Refer to the solution of Tutorial assignment 1. The field due to a single plane carrying a linear current density is in – direction above the plane while below the plane it is in 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.
- Referring to the solution of tutorial problem 1, the field due to the current in xz plane is along above the plane (i.e. for and along below the plane (i.e. for.
y
x
z
O
P
