Question

Suppose there is a steady current in an inductor. If you attempt to reduce the current to zero instantaneously by quickly opening a switch, an arc can appear at the switch contacts. Why? Is it physically possible to stop the current instantaneously? Explain.

Short answer

An arc can appear at the switch contacts due to induction of high emf which leads to insulation breakdown of air causing electric arc, when switch opens quickly. It is not physically possible to stop the current instantaneously since it decays exponentially.

Step-by-Step solution

Step-by-step solution

State Faraday’s law of induction and define insulation breakdown of air

Whenever a conductor is placed in a varying magnetic field, an electromotive force $\left(\epsilon \right)$is induced. Magnetic field can be varying by changing current with time which leads to time rate of magnetic flux $\left({\varphi }_B\right)$

$\epsilon =\frac{d{\varphi }_B}{dt}$

localid="1664163913793" $\epsilon \propto -\frac{dI}{dt}$where $\epsilon$is emf induced and $\frac{dI}{dt}$is time rate of current.

Insulation breakdown of material is the phenomenon when the insulating material fails to prevent flow of current through it under high voltage. The voltage at which breakdown of insulation property of material starts to happen, is called breakdown voltage. When the voltage applied across the material is greater than breakdown voltage, material become conducting, losing its insulation.

When the insulating material is air and voltage is higher than breakdown voltage of air then air loses its insulating property and converts into conducting material by ionizing itself which appears as electric arc. Arc discharge is spark that form in short time at high pressure and high voltage. High pressure at switch since it opens quickly.

Apply Faraday’s law of induction

By quickly opening the switch, current rapidly decreases to zero leading high time rate of current or flux. This high rate induces high value of back emf as per Faraday’s law.

This back emf is greater than breakdown voltage which causes arc discharge.

It is not physically possible to stop the current instantaneously because it decays as per the following equation:

$I=I_{o}e\left({}^{-\frac{R}{Lt}}\right)t$ where $I_0$is current at time $t=0$, $R$is resistance and $L$is inductance.

Therefore, arc can appear at the switch contacts due to induction of high emf which leads to insulation breakdown of air causing electric arc, when switch opens quickly. It is not physically possible to stop the current instantaneously since it decays exponentially.