You are conducting an experiment in which a metal bar of length 6.00 cm and mass 0.200 kg slid

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You are conducting an experiment in which a metal bar of...

You are conducting an experiment in which a metal bar of length $6.00 \mathrm{~cm}$ and mass $0.200 \mathrm{~kg}$ slides without friction on two parallel metal rails (Fig. $\mathbf{P 2 9 . 6 5}$ ). A resistor with resistance $R=0.800 \Omega$ is connected across one end of the rails so that the bar, rails, and resistor form a complete conducting path. The resistances of the rails and of the bar are much less than $R$ and can be ignored. The entire apparatus is in a uniform magnetic field $\vec{B}$ that is directed into the plane of the figure. You give the bar an initial velocity $v=20.0 \mathrm{~cm} / \mathrm{s}$ to the right and then release it, so that the only force on the bar then is the force exerted by the magnetic field. Using high-speed photography, you measure the magnitude of the acceleration of the bar as a function of its speed. Your results are given in the table (next page): $$ \begin{array}{l|rrrrrr} \boldsymbol{v}(\mathbf{c m} / \mathbf{s}) & 20.0 & 16.0 & 14.0 & 12.0 & 10.0 & 8.0 \\ \hline \boldsymbol{a}\left(\mathbf{c m} / \mathbf{s}^{2}\right) & 6.2 & 4.9 & 4.3 & 3.7 & 3.1 & 2.5 \end{array} $$ (a) Plot the data as a graph of $a$ versus $v .$ Explain why the data points plotted this way lie close to a straight line, and determine the slope of the bestfit straight line for the data. (b) Use your graph from part (a) to calculate the magnitude $B$ of the magnetic field. (c) While the bar is moving, which end of the resistor, $a$ or $b$, is at higher potential? (d) How many seconds does it take the speed of the bar to decrease from $20.0 \mathrm{~cm} / \mathrm{s}$ to $10.0 \mathrm{~cm} / \mathrm{s} ?$

You are conducting an experiment in which a metal bar of length $6.00 \mathrm{~cm}$ and mass $0.200 \mathrm{~kg}$ slides without friction on two parallel metal rails (Fig. $\mathbf{P 2 9 . 6 5}$ ). A resistor with resistance $R=0.800 \Omega$ is connected across one end of the rails so that the bar, rails, and resistor form a complete conducting path. The resistances of the rails and of the bar are much less than $R$ and can be ignored. The entire apparatus is in a uniform magnetic field $\vec{B}$ that is directed into the plane of the figure. You give the bar an initial velocity $v=20.0 \mathrm{~cm} / \mathrm{s}$ to the right and then release it, so that the only force on the bar then is the force exerted by the magnetic field. Using high-speed photography, you measure the magnitude of the acceleration of the bar as a function of its speed. Your results are given in the table (next page):
$$
\begin{array}{l|rrrrrr}
\boldsymbol{v}(\mathbf{c m} / \mathbf{s}) & 20.0 & 16.0 & 14.0 & 12.0 & 10.0 & 8.0 \\
\hline \boldsymbol{a}\left(\mathbf{c m} / \mathbf{s}^{2}\right) & 6.2 & 4.9 & 4.3 & 3.7 & 3.1 & 2.5
\end{array}
$$
(a) Plot the data as a graph of $a$ versus $v .$ Explain why the data points plotted this way lie close to a straight line, and determine the slope of the bestfit straight line for the data. (b) Use your graph from part (a) to calculate the magnitude $B$ of the magnetic field. (c) While the bar is moving, which end of the resistor, $a$ or $b$, is at higher potential?
(d) How many seconds does it take the speed of the bar to decrease from $20.0 \mathrm{~cm} / \mathrm{s}$ to $10.0 \mathrm{~cm} / \mathrm{s} ?$

Key Concepts

Lenz’s Law

Lenz’s law states that the direction of the induced current in a conductor is such that its own magnetic field opposes the change in magnetic flux that produced it. This principle not only determines the direction of the induced current but also explains the potential differences that develop in circuits where parts of the conductor are moving, ensuring consistency with the conservation of energy.

Differential Equations for Damping

The linear relationship between the decelerating force (or acceleration) and velocity in electromagnetic damping can be expressed as a first-order differential equation. Solving this differential equation provides insights into how the velocity of the moving conductor decays over time, typically resulting in an exponential decrease. This mathematical framework is essential for predicting time-dependent behavior in systems subject to velocity-proportional damping forces.

Electromagnetic Damping

When a conductor moves in a magnetic field, the induced current produces its own magnetic effects which oppose the motion that created it. This opposition results in a damping force that is proportional to the velocity of the conductor. The phenomenon, sometimes referred to as electromagnetic damping, manifests as a decelerating force on the moving conductor that can be analyzed using Newton’s second law, revealing a linear relationship between acceleration and velocity.

Lorentz Force

A current-carrying conductor in a magnetic field experiences a force known as the Lorentz force. This force, which is calculated as the product of the current, the length of the conductor in the magnetic field, and the magnetic field strength, acts perpendicular to both the direction of the current and the magnetic field. It is the central mechanism by which magnetic fields can exert mechanical influence on moving conductors.

Ohm’s Law in Induced Currents

Ohm’s law describes how the induced current in a circuit is determined by the induced EMF and the circuit’s resistance. In situations where a conductor moves in a magnetic field, the motional EMF drives a current through the circuit and the magnitude of this current is given by dividing the EMF by the resistance. This relationship is critical for relating electromagnetic induction effects to measurable electrical quantities.

Motional EMF

When a conductor moves through a magnetic field, a voltage (electromotive force, or EMF) is induced across its length. This motional EMF, given by the product of the magnetic field strength, the length of the conductor, and its velocity (for motion perpendicular to the field), is a fundamental concept in electromagnetism that forms the basis of electrical generators and is essential in analyzing circuits with moving parts in magnetic fields.

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