The magnetic field on the axis of a dipole ( i.e. at the point P a perpendicular
ID: 2015137 • Letter: T
Question
The magnetic field on the axis of a dipole ( i.e. at the point P a perpendicular distance x from the loop's center ) follows an inverse - cube law: Using the Biot - Savart Law, it can he shown that the magnetic field on axis of a circular loop of radius R carrying a current I is: Using this result, show that very far from the loop ( x > > R ) . Equation 2 reduces to the inverse - cubed law ( i.e. show how to turn Equation 2 into Equation 1 ) . In doing so, identify the magnetic moment mu in terms of the current I and loop radius R. ( Hint: See Figure 4. ) What happens to the magnetic field if you double the distance from the magnet? What if you triple the distance? Again, assume x > > R.Explanation / Answer
a.
you can expand this using using binomial expansion
(x2 + R2)3/2 = can be rewritten as x2(1 + R2/x2)3/2
(1 + R2/x2)3/2 goes to (1)3/2 as R gets much smaller than x.
(you could write out the terms if you wanted, getting rid of anything with a degree higher than 2, it yields the same result)
now we can distribute the x2 back in, giving us (x2)3/2 = x3
the magnetic moment u = IA, = I * R2
b. by doubling the distance, x, the field is reduced by a factor of 8 (1/2)3 = 1/8
by tripling the distance, it decreases by a factor of 27. (1/3)3 = 1/27
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