To find the mechanical power delivered to the rod, we need to analyze the system in terms of electromagnetic induction and the corresponding mechanical forces. Here’s the step-by-step process:

Step 1: Induced EMF (Electromotive Force)

The rod is moving through a magnetic field with velocity $v$, and the induced EMF is given by Faraday's law of electromagnetic induction:

$\mathcal{E} = B \cdot L \cdot v$

where:

• $B = 0.694 \, \text{T}$ (magnetic field strength),
• $L = 0.379 \, \text{m}$ (distance between the rails),
• $v$ is the velocity of the rod.

Step 2: Current in the Circuit

The induced EMF drives a current through the circuit. Using Ohm’s law, the current $I$ in the circuit is:

$I = \frac{\mathcal{E}}{R} = \frac{B \cdot L \cdot v}{R}$

where $R = 12.5 \, \Omega$ is the resistance of the resistor.

Step 3: Magnetic Force on the Rod

The current $I$ flowing through the rod experiences a magnetic force due to the magnetic field. This force $F_{\text{mag}}$ is given by:

$F_{\text{mag}} = I \cdot L \cdot B$

Substitute $I$ from the previous step:

$F_{\text{mag}} = \left( \frac{B \cdot L \cdot v}{R} \right) \cdot L \cdot B = \frac{B^2 \cdot L^2 \cdot v}{R}$

Step 4: Mechanical Power Delivered to the Rod

The mechanical power delivered to the rod is the power associated with the force moving the rod at velocity $v$. This power $P$ is given by:

$P = F_{\text{mag}} \cdot v = \frac{B^2 \cdot L^2 \cdot v^2}{R}$

Step 5: Substitute the Known Values

Now, substitute the known values of $B$, $L$, and $R$:

$P = \frac{(0.694)^2 \cdot (0.379)^2 \cdot v^2}{12.5}$

Simplifying the constants:

$P \approx \frac{0.481636 \cdot 0.143641}{12.5} \cdot v^2 \approx 0.005533 \cdot v^2$

Thus, the mechanical power delivered to the rod is approximately:

$P \approx 0.005533 \cdot v^2$

where $P$ is in watts and $v$ is the velocity of the rod in meters per second (m/s).

Conclusion

The mechanical power delivered to the rod depends on the square of its velocity. If you know the velocity $v$, you can substitute it into the equation to get the exact power.

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Would you like to know more about how to determine $v$ or ask questions about any specific part of this calculation?

5 Related Questions:

1. How can we calculate the velocity of the rod if it's not given in the problem?
2. What are the practical implications of this kind of electromagnetic braking system?
3. How would the power change if the resistance of the resistor were halved?
4. What role does the magnetic field play in determining the current in the circuit?
5. How is the mechanical power related to the energy dissipated by the resistor?

Tip:

Always remember that in induced EMF problems, the relative motion between a conductor and the magnetic field is crucial in determining the induced current and resulting forces.