The current in Fig. E29.18 obeys the equation I(t) = I_0e^(-bt), where b > 0.
Find the direction (clockwise or counterclockwise) of the current induced in the round coil for t > 0.

Verified step by step guidance

1. Identify the direction of the current in the straight wire. The current is flowing downward as indicated by the arrow.
2. Determine the magnetic field created by the straight wire at the location of the round coil. Use the right-hand rule: point your thumb in the direction of the current (downward) and curl your fingers around the wire. Your fingers will curl into the page on the left side of the wire (where the coil is located).
3. Recognize that the magnetic field through the coil is directed into the page and is decreasing over time because the current in the wire is decreasing according to the equation I(t) = I_0e^(-bt).
4. Apply Faraday's Law of Induction, which states that a changing magnetic field through a coil induces an electromotive force (emf) in the coil. The induced emf will create a current in the coil to oppose the change in magnetic flux (Lenz's Law).
5. Since the magnetic field through the coil is decreasing, the induced current in the coil will create its own magnetic field to oppose this decrease. To create a magnetic field into the page, the induced current in the coil must flow in a clockwise direction.

Key Concepts

Electromagnetic Induction

Electromagnetic induction is the process by which a changing magnetic field within a closed loop induces an electromotive force (EMF) and consequently an electric current in that loop. This phenomenon is described by Faraday's law, which states that the induced EMF is proportional to the rate of change of magnetic flux through the loop. In this scenario, the decreasing current in the nearby wire creates a changing magnetic field that affects the coil labeled 'A'.

Lenz's Law

Lenz's Law states that the direction of the induced current in a closed loop will be such that it opposes the change in magnetic flux that produced it. This means that if the magnetic field through the loop is decreasing, the induced current will flow in a direction that attempts to maintain the magnetic field. In this case, as the current in the wire decreases, the induced current in the coil will flow in a direction that opposes this decrease.

Direction of Current Flow

The direction of current flow in a circuit is conventionally defined as the direction positive charge would move, which is opposite to the flow of electrons. In the context of the coil and the wire, the induced current's direction can be determined using the right-hand rule: if you point your thumb in the direction of the magnetic field (from the wire to the coil) and curl your fingers, they will indicate the direction of the induced current. This helps in determining whether the current is clockwise or counterclockwise.