Question

The current in the figure (Figure 1) obeys the equation I(t) = I0e-bt, where b > 0. Find the direction (clockwise or counterclockwise) of the current induced in the round coil for t > 0.

Answer 1

Concepts and reason

The problem deals with the concept of the right hand rule for the magnetic field and direction of current and Lenz’s law as the direction of the current is to be determined from the circular coil.

Fundamentals

The right hand rule determines the direction of the magnetic field or the direction of the current for a current carrying wire if the direction of one thing is known. The thumb of the right hand gives the direction of current and the curl fingers gives the direction of the magnetic field.

According to the Lenz’s law an induced electromotive force (emf) always gives rise to a current whose magnetic field opposes the change in original magnetic flux.

The direction of the current is towards the negative $x$ axis above the circular coil.

The equation of the current from the straight wire is as $I\left( t \right) = {I_o}{e^{ - bt}},b > 0$

Now the current starts out at the value I₀ and decreases asymptotically to zero, so during the period $t > 0$ the current is directed to the left but is decreasing the entire time.

By the right hand rule when the current is directed towards the left, we know the magnetic field that surrounds the wire is directed out of the page in the area where the round coil is located.

Since the current is decreasing towards the left so the magnetic field is also decreasing out of the page.

Since magnetic field is decreasing for the circular coil when $t > 0$ .

Then by Lenz' Law, the induced current in the round coil must be such that its magnetic field will counteract the reduction of the magnetic field; in other words, the induced current must create a magnetic field directed out of the page also so the magnetic field will increase.

So again from the right hand rule as the magnetic field is out of the page for the circular coil so the direction of the current must be counter clockwise.

Ans:

The direction of the current for the circular coil is counter clockwise.