Field Due to Charge in UCM

Electricity and Magnetism Level pending

Consider a point charge of magnitude Q = + 10 Q = +10 moving around a circle of radius R = 1 R=1 with its centre at the origin. The charge moves at a constant speed of V = 2 π V = 2 \pi . What is the Magnitude of the magnetic flux density ∣ B ⃗ ∣ \lvert \vec{B} \rvert at the origin?

Bonus: Compare the formula you obtain with that for the field at the centre of a current-carrying loop.

Note:

  • μ o = 1 \mu_o = 1

  • UCM - Uniform circular motion

  • Just as for the problem based on which this one is inspired, the formula on the first page of this reference would be useful.

Inspiration


The answer is 5.

Karan Chatrath by Karan Chatrath , 27, Netherlands

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1 solution

Steven Chase
Apr 19, 2020

This problem is more insightful than the one which inspired it. The velocity and the displacement vector are at a right angle, so the cross product boils down to a scalar product.

B = μ 0 q 4 π v r 2 B = \frac{\mu_0 q}{4 \pi} \frac{v}{r^2}

Plugging in numbers gives B = 5 B = 5 . Let's suppose that this moving charge is equivalent to a current loop. What is the equivalent current value? Equate the field expression for a circular current loop to the one just derived for the moving charge.

μ 0 I 2 r = μ 0 q 4 π v r 2 I = q v 2 π r \frac{\mu_0 I}{2 r} = \frac{\mu_0 q}{4 \pi} \frac{v}{r^2} \\ I = \frac{q v}{2 \pi r}

Now consider the time period of motion for the charge:

T = 2 π r v T = \frac{2 \pi r }{v}

Substituting in gives:

I = q T I = \frac{q}{T}

In summary, if a point charge q q moves in uniform circular motion with time period T T , it has the same effect as a current loop with I = q T I = \frac{q}{T} . This is a very nice and intuitive result.

1 Helpful 1 Interesting 1 Brilliant 0 Confused

I tried to think of generalising this result. First off, I thought of the result analogous to the field due to a straight wire. There, a single point charge will not produce the analogous known formula. One must think of a wire with linear charge density where each elementary charge moves at a given speed. Essentially, this would result in deriving the Biot-Savart law itself. This really puts this principle into perspective.

Karan Chatrath - 1 year, 1 month ago

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