Answer to Question #289840 in Electric Circuits for boshra

Question

How does the strength of the magnetic field that is 1 cm from a current-carrying 

       wire compare with the strength of the magnetic field that is 4 cm from the wire?

Answer 1

Explanation & Calculations

• The formula for the magnetic field strength at a point $\small r$ far from a straight conductor is

$\qquad\qquad \begin{aligned} \small B&=\small \frac{\mu i}{2\pi r} \end{aligned}$

• Since this question deals with the same wire with the same current in the same medium ($\small \mu$ is constant), this formula can be reduced down to proportional mode as follows,

$\qquad\qquad \begin{aligned} \small B&\propto \frac{1}{r} \end{aligned}$

• Then by comparison, you get,

$\qquad\qquad \begin{aligned} \small B_1&\propto \small \frac{1}{1\,cm}\cdots(1)\\ \\ \small B_4&\propto \small \frac{1}{4\,cm}\cdots(2)\\ \\ (2)\,\div\,(1),\\ \\ \small \frac{B_4}{B_1}&=\small \frac{1}{4}\times\frac{1}{1}\\ \small B_4&=\small \frac{1}{4}.B_1 \end{aligned}$

• Therefore, the field at the far location is one-fourth of that at the near location.