Question

Electromagnetic Induction - Decay of Current in an Inductive Circuit

Answer 1

Electromagnetic induction is the process by which a changing magnetic field induces an electromotive force (EMF) or voltage in a conductor. When a current flows through a conductor, it generates a magnetic field around it. Conversely, when a magnetic field changes around a conductor, it induces a voltage across the conductor. This phenomenon is the basis for the operation of inductive circuits.

In an inductive circuit, such as a coil or an inductor connected in series with a resistor, when the circuit is energized (current starts flowing), a magnetic field builds up around the coil. When the current through the coil changes, either due to a change in the applied voltage or due to the circuit being switched off, the magnetic field around the coil also changes. According to Faraday's law of electromagnetic induction, this change in the magnetic field induces an EMF (voltage) in the coil, which, in turn, opposes the change in current.

The induced EMF creates a phenomenon known as "self-induction," which resists any change in the current flowing through the inductor. As a result, when the circuit is switched off or the voltage is removed, the current does not immediately drop to zero. Instead, the current begins to decrease gradually.

The rate of decay of current in an inductive circuit is determined by the inductance of the coil (measured in Henrys) and the resistance of the circuit. The higher the inductance, the slower the current decay. The higher the resistance, the faster the current decay. The mathematical relationship governing the decay of current in an RL (resistor-inductor) circuit is given by the following differential equation:





=
−



dt
di
    ​

=−
L
R
    ​

i

Where:





dt
di
    ​

 is the rate of change of current with respect to time.

R is the resistance of the circuit.

L is the inductance of the coil.

i is the instantaneous current.

The solution to this equation is an exponential decay of the current with time:


(

)
=

0
⋅

−



i(t)=i
0
    ​

⋅e
−
L
R
    ​

t

Where:


(

)
i(t) is the current at time

t.

0
i
0
    ​

 is the initial current when the circuit is switched off (at

=
0
t=0).

e is the base of the natural logarithm.

This equation shows that as time (

t) increases, the current (

(

)
i(t)) exponentially approaches zero. The time constant

τ of the decay is given by

=


τ=
R
L
    ​

. This time constant represents the time it takes for the current to decrease to approximately 36.8% of its initial value.

In summary, the decay of current in an inductive circuit is a result of the self-induction effect, which opposes changes in current. The rate of decay depends on the inductance and resistance of the circuit, and the current follows an exponential decay pattern over time.