A corner point makes angles θ_{1} and θ_{2} 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 = (μ_{0}/4π)*(i*ds*sin(θ)/r^{2})

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/sin^{2}(θ) = a*dθ/sin^{2}(θ) (3)

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

∫dB = I*(μ_{0}/4π)∫dx*sin(θ)/r^{2} (4)

We can use the equations (1) and (3) for the expression dx/r^{2} and we get

dx/r^{2} = dx*sin^{2}(θ)/a^{2} = a*(dθ/sin^{2}(θ))*sin^{2}(θ)/a^{2} = dθ/a

or

dx/r^{2} = dθ/a (5)

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

∫dB = I*(μ_{0}/(4πa))∫sin(θ)dθ (6)

To find the magnitude of the total magnetic field at the point, we need to integrate from θ=θ_{1} to θ=θ_{2}.

Thereby,

B = (I*(μ_{0}/(4πa))*(cos(θ_{1})-cos(θ_{2})).

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.