PHYS208 Fundamentals of Physics II

Quiz 7 -- Current Distributions

Field lines of circular current ring

Evaluate the magnetic field B on the axis of a circular current loop at point P, a distance x from the center of the loop having radius a.

The magnetic field lines are shown at right. There is not enough symmetry in this situation to invoke Ampere’s law easily for the calculation of the field.

Solution:

  1. What is the formula for the Biot-Savart law?

    [Biot-Savart law]

    Contribution to dB

  2. What is the direction of the contribution to the field dB contributed by the infinitesimal current segment ds1 shown at the top of the loop? Show it on the sketch.

  3. Consider the conjugate element ds2 at the bottom of the loop. Present a sketch or brief discussion that demonstrates that the total field B at point P is horizontal.

    Two contributions to dB

    Since the final effect of each dB is that only its horizontal component contributes to the total b, our final integral need involve only that component from the Biot-Savart law. Any infinitesimal segment ds contributes the same horizontal component regardless of its position on the loop.

  4. Express the horizontal component of dB in terms of x and a.

    Since ds is perperdicular to r for all differential current elements, regardless of their position on the loop:

    cross product magnitude

    Thus the magnitude of the contribution to the magnetic field from the current segment of arc length ds is:

    dB magnitude

    Horizontal projection of dB

    dB magnitude

  5. What is the total field strength B at point P?

    Integrating over all current segments reduces to evaluation of the circumference of the loop since all contributions dB have the same magnitude:

    total B

  6. Check the limit of the above expression for x = 0, the point at the center of the loop.

    The denominator involving x and a above reduces to a3 in this limit. Thus magnetic field at the center of the loop is

    B at center of loop

    as expected.

  7. Check the limit far away; i.e., x>>a.

    The denominator involving x and a above reduces to x3 in this limit. Thus the magnetic field behaves as a dipolar field, as expected!

    B at center of loop


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Last updated April 17, 1998.
Copyright George Watson, Univ. of Delaware, 1998.