A wheel with 20 metallic spokes, each 1 m long, is rotated with a speed of 120 rpm in a plane perpendicular to

Question

A wheel with 20 metallic spokes, each 1 m long, is rotated with a speed of 120 rpm in a plane perpendicular to a magnetic field of 0.4 G. The induced emf between the axle and rim of the wheel will be: (1 G = 10⁻⁴ T)

Accepted Answer

1.  **Formula:** The induced electromotive force (emf) $\varepsilon$ between the center (axle) and the rim of a rotating rod (spoke) is given by $\varepsilon = \frac{1}{2} B \omega R^2$, where $B$ is the magnetic field, $\omega$ is the angular velocity, and $R$ is the length of the spoke .
2.  **Parallel Combination:** Although there are 20 spokes, they are all connected in parallel between the axle and the rim. In a parallel combination of identical cells (spokes), the equivalent emf is equal to the emf of a single cell. Therefore, the number of spokes does not affect the magnitude of the induced voltage .
3.  **Unit Conversions:**
    *   Magnetic Field ($B$): $0.4 	ext{ G} = 0.4 	imes 10^{-4} 	ext{ T}$.
    *   Angular Velocity ($\omega$): $120 	ext{ rpm} = \frac{120 	imes 2\pi}{60} 	ext{ rad/s} = 4\pi 	ext{ rad/s}$.
    *   Length ($R$): $1 	ext{ m}$.
4.  **Calculation:**
    *   $\varepsilon = \frac{1}{2} 	imes (0.4 	imes 10^{-4}) 	imes (4\pi) 	imes (1)^2$
    *   $\varepsilon = 0.2 	imes 10^{-4} 	imes 4 	imes 3.14$
    *   $\varepsilon = 0.8 	imes 3.14 	imes 10^{-4}$
    *   $\varepsilon \approx 2.512 	imes 10^{-4} 	ext{ V}$.

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