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Questions
Derive the expression for magnetic field produced by a current in a circular arc of wire.
Derive an expression for the magnetic field produced by a current in a circular arc of a wire using Biot-Savart law.
Derive an expression for the magnetic field produced at the centre of a circular arc of a wire carrying current I.
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Solution
Consider a circular arc of a wire (XY), carrying a current I.
The circular arc XY subtends an angle θ at the centre O of the circle with radius r, of which the arc is a part, as shown in the figure below.
Current-carrying wire of a shape of the circular arc.
Consider length element d`vec l` lying always perpendicular to `vec r`.
Using Biot-Savart law, the magnetic field produced at O is:
`d vec B = mu_0/(4 pi) (I d vec l xx vec r)/r^3`
dB = `mu_0/(4 pi) I (d l r sin 90^circ)/r^3`
= `mu_0/(4 pi) (I d l)/r^2` ...(1)
Equation (1) gives the magnitude of the field. The direction of the field is given by the right-hand rule.
Thus, the direction of each of the dB is into the plane of the paper. The total field at O is:
B = `int dB`
= `mu_0/(4 pi) I int_A^theta (d l)/r^2`
= `mu_0/(4 pi) I int_A^theta (r d theta)/r^2`
= `mu_0/(4 pi) I/r theta` ...(2)
Where the angle θ is in radians.
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