Magnetic field due to a current in a straight wire

BiotSavartImg1A corner point makes angles θ1 and θ2 with the ends of a straight wire carrying current I. The distance of the point from the wire is a. What is the
magnetic field at the point?

Solution.

The magnitude of the field produced at point at distance r by a current-length element i turns out to be

dB = (μ0/4π)*(i*ds*sin(θ)/r2)

where θ is the angle between the directions of vectors ds and r. This equation is known as the law of Biot and Savart.

We can see that

a = r*sin(180-θ) = r*sin(θ) or r = a/sin(θ)     (1)

x = r*cos(180-θ) = –r*cos(θ) or x = -a*cot(θ)     (2)

So,

dx = -a*dcot(θ) = -a*-dθ/sin2(θ) = a*dθ/sin2(θ) (3)

We can find the magnitude of the magnetic field produced at the corner point by integrating dB.

∫dB = I*(μ0/4π)∫dx*sin(θ)/r2 (4)

We can use the equations (1) and (3) for the expression dx/r2 and we get

dx/r2 = dx*sin2(θ)/a2 = a*(dθ/sin2(θ))*sin2(θ)/a2 = dθ/a

or

dx/r2 = dθ/a (5)

Now, we substitute the expression (5) into the equation (4) and we get

∫dB = I*(μ0/(4πa))∫sin(θ)dθ (6)

To find the magnitude of the total magnetic field at the point, we need to integrate from θ=θ1 to θ=θ2.

Thereby,

B = (I*(μ0/(4πa))*(cos(θ1)-cos(θ2)).

Using the right-hand rule we can get that the magnetic field (at the point) is directed out of the page, as indicated by the dot.

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