A current 1 is flowing through the sides of an equilateral triangle of side a. The magnitude of the magnetic field at the centroid of the triangle is A)9μ0I B)μ0I/πa C)3μ0I/2πa D)3μ0I/πa( Answer is option a) can you give solution & explanation?

Question

Answer

To find the magnitude of the magnetic field at the centroid of the equilateral triangle, we can use Ampere's Law. Ampere's Law states that the line integral of the magnetic field around a closed loop is equal to the product of the current enclosed by the loop and the permeability of free space (μ0).

In this case, we can choose a circular loop centered at the centroid of the triangle with radius r. The circumference of the loop is equal to the sum of the three sides of the equilateral triangle, which is 3a. The current enclosed by the loop is the total current flowing through the sides of the triangle, which is I.

Using Ampere's Law, we have:

∮ B · dl = μ0Ienc

where B is the magnetic field, dl is an infinitesimal element of the loop, and Ienc is the current enclosed by the loop.

Since the magnetic field is constant along the loop, we can take it out of the integral:

B ∮ dl = μ0Ienc

The left-hand side of the equation is the product of the magnetic field and the circumference of the loop, which is 2πr. Therefore, we have:

B (2πr) = μ0I

Solving for B, we get:

B = μ0I / (2πr)

Since the centroid of the triangle is equidistant from all three sides, the radius r is equal to the height of the equilateral triangle, which can be calculated as h = (√3/2) * a.

Substituting this value into the equation, we have:

B = μ0I / (2π(√3/2) * a)
B = μ0I / (π√3a)

Therefore, the magnitude of the magnetic field at the centroid of the triangle is μ0I / (π√3a).

Comparing this with the given options, we can see that the correct answer is option B) μ0I / (πa).

Similar Physics Doubts

A current of 1 ampere is flowing in the sides of an equilateral triangle of side 4.5 x 10-2m,then the magnetic field at the centroid of the triangle (inμT) is ____. Correct answer is '40'. Can you explain this answer?

At an instant shown, three point masses m, 2m and 3m rest on a horizontal surface, and are at the vertices of an equilateral triangle of unit side length. Assuming that G is the gravitational constant, the magnitude and direction of the torque on the mass 3m, about the point O, at that instant is

Letz1,and z2 be non-zero complex numbers satisfyingThen the triangle made by points with vertices at origin,z1 and z2 is

Three equal chargesQare placed at the three vertices of an equilateral triangle. What should be the value of a charge, then when placed at the centroid, reduces the interaction energy of the system is zero

Top Courses for Physics

Suggested Free Tests

Related Content

Schools

Competitive Examinations

Quick Access Links

Technical Exams