Question

Write the expression for the magnetic field due to a current element in vector form. Consider a 1 cm segment of a wire, centered at the origin, carrying a current of 10 A in positive x-direction. Calculate the magnetic field `vec B` at a point (1 m, 1 m, 0).

Answer 1

Given: Length of segment (dl) = 1 cm = 10⁻² m

Current (I) = 10 A

Segment along +x direction (dl) = `10^-2 hat i`

Field point = (1, 1, 0)

Position vector  (r) = `hat i + hat j`

= `sqrt(1^2 + 1^2)`

= `sqrt2`

dl × r = `(10^-2 hat i) xx (hat i + hat j)`

Using cross products:

`hat i xx hat i` = 0, `hat i xx hat j = hat k`

dl × r = `10^-2 hat k`

Magnetic field due to a current element is given by the Biot–Savart law:

dB = `mu_0/(4 pi) (I dl xx hat r)/r^2`

Vector expression (Biot–Savart law):

dB = `mu_0/(4 pi) (I(dl xx r))/r^3`

= `mu_0/(4 pi) (I(10^-2 hat k))/((sqrt2)^3)`

= `(10^-7 xx 10 xx 10^-2)/(2 sqrt 2) hat k`

B = `10^-8/(2 sqrt2) hat k`

= `10^-8/2.828 hat k`

B ≈ `3.5 xx 10^-9 hat k T`