Question

Two geometrically identical cycle wheels have different number of spokes connecting centre to rim. One is having 20 spokes and other having only 10 (the rim and the spokes are resistanceless). A resistance of value \(\mathrm{R}\) is connected between centre and rim in both the wheels. If same magnetic field is applied perpendicular to the plane of these wheels and both wheels are rotating with the same angular speed, then the current in \(\mathrm{R}\) will be : (1) double in first wheel than in the second wheel (2) four times in first wheel than in the second wheel (3) will be double in second wheel than that of the first wheel (4) will be equal in both these wheels

Short answer

The current will be equal in both wheels.

Step-by-step solution

- Understand the Role of the Spokes

Since the spokes are geometrically identical and resistanceless, their effect on the resistance is negligible, meaning the resistance in each wheel setup merely depends on the resistor R connected between the center and the rim.

- Analyze the Magnetic Flux

When a magnetic field is applied perpendicular to the plane of the wheels, it induces an electromotive force (EMF) in the rim.

- Calculate the EMF

The electromotive force (EMF, \emf) induced depends on the rate of change of the magnetic flux through the area of the wheel. Since both wheels are rotating with the same angular speed \(\boldsymbol{\theta}\) in identical magnetic fields, the EMF induced in both wheels will be identical.

- Determine the Current

The current \(I\backslash backslash \) in the resistor R can be found using Ohm’s law. Since EMF = IR, and the EMF is the same for both wheels, the current will be same for both resistors.

- Conclusion

Therefore, the number of spokes does not affect the EMF or the current through R. The current in resistor R will be equal for both wheels.

Key concepts

magnetic induction

Magnetic induction is a fundamental concept in electromagnetism. It refers to the process by which a change in magnetic field within a closed loop of wire induces an electromotive force (EMF). This principle is known as Faraday's Law of Induction and is mathematically expressed as:
\[ \text{EMF} = -\frac{d\Phi_B}{dt} \]
where \( \Phi_B \) is the magnetic flux, which is the product of the magnetic field (\

Ohm's Law

Ohm's Law is a key principle in the study of electric circuits. It describes the relationship between voltage (V), current (I), and resistance (R) in a circuit. It is mathematically represented as:
\[ V = IR \]
In the context of the given problem, we can substitute EMF (\text{ircle{E}}) for voltage because the EMF induced in the wheel acts like a voltage source. Thus, the current (I) in the resistor R can be found using Ohm's Law:
\[ \text{EMF} = IR \]
Given that the EMF is the same for both wheels, and the resistance R is identical in both setups, the current through the resistor will be the same for both wheels, regardless of the number of spokes.

electromotive force

Electromotive force (EMF) is a crucial concept to understand in this problem. EMF is generated when a conductor moves through a magnetic field or when the magnetic field around a conductor changes. It is defined as the energy provided per unit charge to move the charge through the circuit. The unit for EMF is voltage (V).

The induced EMF in a rotating wheel can be calculated using the formula for a rotating conductor in a magnetic field:

\[ \text{EMF} = \text{B} \times \text{l} \times \text{v} \]
where \= xml{1}{\range}{B} is the magnetic field strength, \text{l} is the length of the conductor (or the radius of the wheel from center to rim), and \text{v} is the tangential velocity. Because both wheels are rotating with the same angular speed (\theta), and are in the same uniform magnetic field, the induced EMF will be the same for both wheels.

This concept highlights that the number of spokes, being geometrically identical and resistanceless, do not affect the EMF induced in the wheels. Therefore, in both cases, the EMF is the same. This understanding strengthens our conclusion about the current being equal in both wheels when passed through the same resistor.