Two parallel conducting rails with negligible resistance are...

Two parallel conducting rails with negligible resistance are connected at one end by a resistor of resistance $R$, as shown in the figure. The rails are placed in a magnetic field $\vec{B}_{\text {ext }}$, which is perpendicular to the plane of the rails. This magnetic field is uniform and time independent. The distance between the rails is $\ell$. A conducting rod slides without friction on top of the two rails at constant velocity $\vec{v}$. a) Using Faraday's Law of Induction, calculate the magnitude of the potential difference induced in the moving rod. b) Calculate the magnitude of the induced current in the rod, $i_{\text {ind }}$ c) Show that for the rod to move at a constant velocity as shown, it must be pulled with an external force, $\vec{F}_{\text {ext }}$ and calculate the magnitude of this force. d) Calculate the work done, $W_{\text {ext }},$ and the power generated, $P_{\text {ext }}$, by the external force in moving the rod. e) Calculate the power used (dissipated) by the resistor, $P_{\mathrm{R}}$. Explain the correlation between this result and those of part (d).

Two parallel conducting rails with negligible resistance are connected at one end by a resistor of resistance $R$, as shown in the figure. The rails are placed in a magnetic field $\vec{B}_{\text {ext }}$, which is perpendicular to the plane of the rails. This magnetic field is uniform and time independent. The distance between the rails is $\ell$. A conducting rod slides without friction on top of the two rails at constant velocity $\vec{v}$.
a) Using Faraday's Law of Induction, calculate the magnitude of the potential difference induced in the moving rod.
b) Calculate the magnitude of the induced current in the rod, $i_{\text {ind }}$
c) Show that for the rod to move at a constant velocity as shown, it must be pulled with an external force, $\vec{F}_{\text {ext }}$ and calculate the magnitude of this force.
d) Calculate the work done, $W_{\text {ext }},$ and the power generated, $P_{\text {ext }}$, by the external force in moving the rod.
e) Calculate the power used (dissipated) by the resistor, $P_{\mathrm{R}}$. Explain the correlation between this result and those of part (d).

Key Concepts

Faraday's Law of Electromagnetic Induction

Faraday’s law states that a time-varying magnetic flux through a closed loop induces an electromotive force (emf) in that loop. This concept is used to determine the magnitude of the induced potential difference when a conductor moves in a magnetic field, thereby changing the flux linkage through the circuit.

Motional EMF

Motional emf refers to the voltage generated across a conductor moving through a magnetic field. It arises because the charges inside the rod experience a magnetic force which separates them, resulting in an induced potential difference proportional to the product of the magnetic field strength, the length of the conductor, and its velocity.

Ohm's Law

Ohm's law relates the current flowing through a resistor to the voltage across it and the resistance it provides. In the context of induced emf, it is used to calculate the induced current by dividing the induced voltage by the resistance encountered in the circuit.

Lorentz Force and Magnetic Drag

The Lorentz force describes the force acting on a moving charge in a magnetic field, which, when summed over a current-carrying conductor, results in a magnetic force opposing the motion of the conducting rod. This magnetic drag must be countered by an external force if the rod is to maintain constant velocity.

Energy Conservation and Power Dissipation

Energy conservation in electromagnetic systems connects the work done by an external force to the electrical energy generated and eventually dissipated. The mechanical work that must be done against the magnetic drag appears as electrical energy, which is either stored or dissipated as heat in the resistor, reflected in power calculations.

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