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Faraday's Law of Induction is fundamental to understanding how a changing magnetic environment can generate an electric current. This principle is at the core of many devices like generators and transformers. In this context, when the conducting rod moves through a magnetic field, it experiences a change in magnetic flux. This change induces an electromotive force (EMF) in the rod. The formula to calculate the EMF is given by: \[ \text{EMF} = B \cdot L \cdot v \]where

• \(B\) is the magnetic field strength in tesla (T),
• \(L\) is the length of the rod in meters (m),
• \(v\) is the velocity of the rod in meters per second (m/s).

This relationship demonstrates that the induced voltage (EMF) is dependent on how fast the magnetic field is cut, the strength of the magnetic field, and the length of the rod. As the rod in our problem starts moving, the velocity increases, thus causing an increase in the induced EMF. Hence, as per Faraday's Law, the faster the rod moves, the greater the induced EMF.

Lenz's Law is an extension of the principles explained by Faraday's Law. It predicts the direction of the induced current. According to Lenz's Law, the induced current creates a magnetic field that opposes the change in the original magnetic field. This is expressed in saying "induced currents counter the change that produced them." In our scenario with the rod, the motion through the magnetic field generates a current in a direction that opposes the motion of the rod.
Here's how it works:

• As the rod moves to the right, an upward force acts on it due to the induced current.
• This creates a magnetic field that opposes the motion of the rod.
• The result is a damping force that tries to slow down the rod's movement.

This opposing induced force is why it's sometimes challenging to keep objects like this rod moving through a magnetic field; the induced fields respond to resist any change in motion.

Ohm's Law is a staple in the study of electricity and magnetism, describing the relationship between current, voltage, and resistance. It is succinctly expressed as \[ I = \frac{\text{EMF}}{R} \]where

• \(I\) is the current in amperes (A),
• \(\text{EMF}\) is the electromotive force in volts (V),
• \(R\) is the resistance in ohms (Ω).

In the exercise, once the EMF has been induced by the rod's motion in the magnetic field, Ohm's Law allows us to calculate the induced current through the rod. Since the resistance in the rod is constant (80.0 Ω), the induced current depends directly on the EMF. Hence, a greater velocity not only increases the EMF but also proportionally increases the current flowing through the rod. This relationship is crucial to solve for the damping force as part of the motion equation.

Newton's Second Law of Motion, typically referred to as \( F = m \cdot a \), expresses that the force acting on an object is equal to the mass of the object multiplied by its acceleration.
In the context of the exercise, the rod is subjected to two main forces:

• The applied force \(F\), which aims to move the rod to the right.
• The magnetic damping force \(F_{\text{damping}}\), which opposes its motion.

The net force acting on the rod is thus the difference: \[ F_{\text{net}} = F - F_{\text{damping}} \]Using Newton's Second Law, we write \[ m \cdot a = F_{\text{net}} \]This relationship allows us to solve for the acceleration \(a\), knowing the masses and forces involved. Finally, to find the time \(t\) when the rod reaches a specified speed, we rearrange the equation to \[ t = \frac{v_f}{a} \]where \(v_f\) is the final velocity. This connection helps us understand how the interplay of forces influences the motion of the rod over time.