Question

A $2.0cm\times 2.0cm$square loop of wire with resistance$0.010\Omega$has one edge parallel to a long straight wire. The near edge of the loop is cm from the wire. The current in the wire is increasing at the rate of $100A/s$. What is the current in the loop?

Short answer

The current in loop,$I_{\text{induced}}=44.2\mu \mathrm{A}$

Step-by-step solution

Introduction of loop

A loop is an electromagnetic system that uses two wires to send signals to a device. It can be used to produce energy or transmit data.

Simplify by values

Given values:

$A=2.0\mathrm{cm}\times 2.0\mathrm{cm}$

$R=0.010\Omega$(ohm )

$d=1.0\mathrm{cm}$

Magnetic flux of wire at distance :

$B=\frac{{\mu }_{o}I}{2\pi x}$

For a small strip of width $dx$flux:

$d\varphi =dAB=\frac{{\mu }_{o}I}{2\pi x}(0.02dx)$

$\varphi ={\int }_{0.01}^{0.03}\frac{{\mu }_{o}I}{2\pi x}dx=\frac{{\mu }_{o}I}{2\pi }\ln 3(0.02)$

Find the worth of induce current

$\epsilon =\frac{d\varphi }{dt}·I_{\text{induced}}=\frac{\epsilon }{R}$
$\Rightarrow I_{\text{induced}}=\frac{{\mu }_o}{2\pi R}\ln 3\frac{dI}{dt}\times (0.02)$

localid="1650218054027">⇒Iinduced=4π×10×(1.098)m2π×0.1Ω×100×(0.02)

$\Rightarrow I_{\text{induced}}=44.2\mu \mathrm{A}$