Physics II: Chapter 7

ReferenceBeginning Physics II




Magnetic Field (Source), Field Produced By Currents, Solenoid, Magnetic Field Lines, Composite Fields, Ampere’s Law, Coaxial Cable



For details on the following concepts, please consult Chapter 7.

One basic origin of a magnetic field is a moving charge or an equivalent current in a wire. It exerts a force on another moving charge. Another basic source for a magnetic field is an electric field that varies with time.

The following is the formula for the magnitude of the field produced by a moving charge q, moving with velocity v, at a point located at a displacement r from the charge.

This is known as the Law of Biot and Savart. The field is zero if φ is zero. The largest magnetic field is produced when φ is 90°. The magnitude of B decreases as 1/r2 with the distance from point a. The field increases with both q and v.

The direction of the field is perpendicular to the plane containing both v and r, and according to the right-hand rule.

If one traces the magnetic field lines, they form concentric circles around the direction of v.

Two charges of the same sign moving parallel to each other in the same direction shall experience magnetic attraction.

Current flowing in a wire is equivalent to moving charge: qv = I∆L. The magnitude of the magnetic field produced by a current I flowing in a wire of length ∆L at a point located at a distance r from the element is,

Field at the Center of a Current Carrying Ring

A current I, flowing in the clockwise direction. Then the field at the center of the ring, with direction into the paper is,

For N turns of the coil the field is multiplied by N.

Field Along the Axis of a Ring

The field at the center of the ring, with direction at P to the right is,

Field of a Long Straight Wire

I is the current in the wire and R is the perpendicular distance of the point P from the wire. The direction of the field is into the paper at P.

A solenoid
is a wire is continuously wound around a long pipe with adjacent windings close to each other. The field within a long solenoid is the same at any point within the solenoid and is zero (or very small) outside the solenoid.

where n is the number of turns per meter. The direction of the field is parallel to the axis with the direction given by the same right-hand rule used for the ring. The result is the reason that solenoids are so very useful for producing magnetic fields. The field produced is uniform, with the same magnitude and direction everywhere within the solenoid. Furthermore, this uniform field does not depend on the radius of the solenoid, only on the number of windings per unit length. One can, for instance, wind several layers of turns, one on top of the other, to increase n, and each layer will contribute the same field, independent of the radius (as long as the solenoid is truly long).

If the solenoid is not infinitely long, but the length is much greater than the radius, then the above result is still nearly true as long as one is not too near to the end of the windings. The field lines inside the solenoid are straight lines, parallel to the axis, until one approaches the ends. Outside the solenoid, the field is no longer zero, and the field lines are as shown above. This happens to be the same field line configuration as for a permanent bar magnet.

The magnetic field lines form closed loops, unlike electric field lines that begin or end at a point charge. The fact that magnetic field lines don’t converge to or diverge from a point is a fundamental property of the magnetic field and can be stated as a general law: “Magnetic field lines never converge to a point or diverge from a point”.

If several wires each produce magnetic fields, then the actual magnetic field at any point is the vector sum of the fields produced by each wire.

There is a powerful general law relating the magnetic field and the current, which often gives insight into the behavior of the magnetic field, and, in certain circumstances, allows for the compete determination of the field without lengthy calculation. This relationship is given by Ampere’s law.

The line integral of B cos θ ∆L around a closed path equals the total current flowing through the area enclosed by that path. This very important result is Ampere’s law.

A coaxial cable consists of an inner solid conductor of radius R1, carrying current I out of the paper, and an outer, concentric hollow cylinder, of radius R2, carrying the same current I into the paper, the current being distributed uniformly around the cylinder.

Since the two currents are equal and flow in opposite directions, the field outside the cable is zero.


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