Magnetism in ring

Electricity and Magnetism Level 3

A uniform circular ring conductor is connected to a battery, as shown.

Here the radius of the ring is 5 5 meters and the current I I is 5 5 amperes. If ∠ A = π 8 , \angle A = \frac{\pi}{8}, find the resulting magnetic field intensity at the center of the ring due to the conductor.

Enter your answer to the nearest integer in microteslas.


The answer is 0.

Ashish Menon by Ashish Menon , 20, India

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1 solution

Ashish Menon
May 30, 2018

Relevant wiki: Magnetic Effects of Current

Since the conductor is uniform, the resistance is uniformly divided along the angle. Since the current divides in inverse ratio of resistance in parallel connection, the current distribution would be as shown below:

Now using the formula for the magnetic field intensity at the center of a circular arc = μ 0 I 2 R × θ 2 π \dfrac{\mu_{0} I}{2 R} \times \dfrac{\theta}{2\pi} ,

  • the magnetic field intensity due to the larger arc = μ 0 2 R × I ( A ) 2 π × ( 2 π − A ) 2 π \dfrac{\mu_{0} }{2 R} \times \dfrac{I(A)}{2\pi} \times \dfrac{(2\pi - A)}{2\pi}
  • the magnetic field intensity due to the smaller arc = μ 0 2 R × I ( 2 π − A ) 2 π × ( A ) 2 π \dfrac{\mu_{0} }{2 R} \times \dfrac{I(2\pi - A)}{2\pi} \times \dfrac{(A)}{2\pi}

Observe that the magnetic field in both cases are equal in magnitude. Now, the direction of magnetic field due to larger arc is outside the plane of the screen (towards us) while that due to the smaller arc is inside the plane of the screen (away from us) by right hand thumb rule. Thus, they get cancelled out and the effective magnetic field intensity at the center of the ring is zero \boxed{\text{zero}} T.

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Nice problem. But the answer is obviously not going to be an integer number of Teslas. So it would be easy to guess that the answer must be zero. To make the problem hack-proof, you could ask for the answer to the nearest micro-Tesla. For the record, I solved it legitimately :)

Steven Chase - 3 years ago

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haha, I didnt think about that, thanks, updated it as per your suggestion :)

Ashish Menon - 3 years ago

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