Problem 33 - Physics for Scientists and Engineers with Modern Physics

Question

(II) A conducting rod rests on two long frictionless parallel rails in a magnetic field \(\overrightarrow{\mathbf{B}}(\perp\) to the rails and rod \()\) as in Fig. \(29-44 .(a)\) If the rails are horizontal and the rod is given an initial push, will the rod travel at constant speed even though a magnetic field is present? (b) Suppose at \(t=0\) when the rod has speed \(v=v_{0},\) the two rails are connected electrically by a wire from point a to point b. Assuming the rod has resistance \(R\) and the rails have negligible resistance, determine the speed of the rod as a function of time. Discuss your answer.

Step-by-Step Solution

Answer

(a) No, the rod slows down due to magnetic damping. (b) Velocity decreases exponentially: \( v(t) = v_0 e^{-\frac{B^2 l^2 t}{mR}} \).

Step 1: Analyze the Forces on the Rod

The rod is subject to a magnetic field perpendicular to its motion. Due to this, when it moves, an electromotive force (EMF) is induced in the rod according to Faraday's Law. This EMF will cause a current to flow in the closed circuit formed by the rod, rails, and connecting wire. This, in turn, results in a magnetic force, given by Lorentz force law, acting opposite to the rod's initial direction of motion, slowing it down.

Step 2: Formulate the Expression for the EMF

The EMF ( \( \mathcal{E} \) ) induced is given by Faraday’s law. If the length of the rod is \( l \) , and \( v \) is the velocity of the rod at any time \( t \), the EMF is \( \mathcal{E} = B \cdot l \cdot v \), where \( B \) is the magnetic flux density.

Step 3: Determine the Current in the Circuit

The current \( I \) flowing in the circuit can be calculated using Ohm's Law, \( I = \frac{\mathcal{E}}{R} = \frac{B \cdot l \cdot v}{R} \).

Step 4: Calculate the Magnetic Force on the Rod

The magnetic force \( F_B \) on the rod, due to the current, is given by \( F_B = I \cdot l \cdot B = \frac{B^2 \cdot l^2 \cdot v}{R} \). This force acts in the opposite direction of the motion, providing a decelerating effect.

Step 5: Set Up and Solve the Differential Equation for Velocity

The force \( F_B \) results in a deceleration. Using Newton's second law, \( m \cdot a = -F_B \) , where \( m \) is the mass of the rod, leads to \( m \cdot \frac{dv}{dt} = -\frac{B^2 \cdot l^2 \cdot v}{R} \). Rearrange to \( \frac{dv}{v} = -\frac{B^2 \cdot l^2}{mR} \cdot dt \) and integrate \( v \) from \( v_{0} \) to \( v(t) \) and \( t \) from 0 to \( t \).

Step 6: Integrate the Equation to Find Velocity as a Function of Time

Integrate both sides to find \( \ln(v)-\ln(v_0)= -\frac{B^2 \cdot l^2 \, t}{mR} \). Simplifying gives \( v(t) = v_0 \cdot e^{-\frac{B^2 \cdot l^2 \, t}{mR}} \). This shows that \( v(t) \) decreases exponentially over time.

Step 7: Discuss the Result

The rod will not travel at a constant speed unless external energy is added because the magnetic force acts as a resisting force, creating a damping effect that slows the rod down over time. As time progresses, the rod's velocity decreases exponentially, eventually coming to a stop.

Key Concepts

Lorentz force

When it comes to the behavior of charged particles, the Lorentz force plays a crucial role. It governs how conductors move within a magnetic field. In the context of our exercise, the moment the conducting rod, which has free charges, moves through the magnetic field, it experiences a force. This force acts perpendicular to both the velocity of the rod and the magnetic field direction. The formula for this force is given by:

• \[ \mathbf{F} = q(\mathbf{v} \times \mathbf{B}) \]

where:

• \( q \) is the charge,
• \( \mathbf{v} \) is the velocity of the charge,
• \( \mathbf{B} \) is the magnetic field.

For the rod, this force translates into what's known as the electromagnetic damping force. Its direction is opposite to the rod’s velocity, acting as a brake that slows it down. Understanding how the Lorentz force works helps us appreciate why the rod doesn’t maintain its speed without intervention.

Faraday's Law

Faraday's Law describes how changing magnetic fields can induce electric currents. In our scenario, as the rod moves along the rails, it carries with it a "cutting" action through the magnetic field lines. This movement leads to a change in the magnetic flux through the circuit formed by the rod and rails, thus inducing an electromotive force (EMF). According to the law:

• \[ \mathcal{E} = B \cdot l \cdot v \]

where:

• \( \mathcal{E} \) is the induced EMF,
• \( B \) is the magnetic field strength,
• \( l \) is the length of the rod,
• \( v \) is the velocity of the rod.

This EMF generates currents within the circuit, setting the stage for forces we see in the context of the Lorentz force to play out. Faraday's Law is pivotal in understanding electromagnetic induction and its practical applications.

Ohm's Law

Ohm's Law is a fundamental principle in understanding electrical circuits. Within this exercise, it allows us to relate the induced EMF to the current flowing through the rod, rails, and connecting wire circuit. The law states:

• \[ I = \frac{\mathcal{E}}{R} \]

where:

• \( I \) is the current,
• \( \mathcal{E} \) is the electromotive force,
• \( R \) is the resistance of the rod.

With this relationship, we discern how much current the induced EMF can drive through the circuit. This current is directly responsible for generating the magnetic force that opposes rod motion. Understanding Ohm’s Law allows students to see the linkage between voltage, resistance, and current in any given circuit.

Exponential decay

The concept of exponential decay is crucial in explaining how the speed of the rod diminishes over time. When forces, like the decelerating magnetic force here, act in opposition to motion, they can lead to an exponential decrease in velocity. The mathematical expression captures this behavior as:

• \[ v(t) = v_0 \cdot e^{-\frac{B^2 \cdot l^2 \cdot t}{mR}} \]

where:

• \( v(t) \) is the velocity at time \( t \),
• \( v_0 \) is the initial velocity,
• \( B \), \( l \), \( m \), and \( R \) are constants relating to the system's properties.

This formula reveals how, over time, the rod's speed decreases to zero. Exponential decay underscores why the rod, unless continuously powered, won't maintain a constant speed indefinitely.

Newton's second law

Newton’s Second Law provides the foundational framework for comprehending motion and the effects of forces. Its core is the relationship between force, mass, and acceleration, formulated as:

• \[ m \cdot a = \sum F \]

In our exercise involving the rod, it is rewritten to include the decelerative magnetic force:

• \[ m \cdot \frac{dv}{dt} = -\frac{B^2 \cdot l^2 \cdot v}{R} \]

Here:

• \( m \) is the mass of the rod,
• \( a \) is the acceleration,
• the negative sign indicates the force is opposing the velocity direction,
• over time, as the force acts, velocity reduces and can be eventually reduced to zero.

It tells us how the rod’s velocity changes over time under the influence of the magnetic drag force.