Question 17: Electromagnetic Induction

Question

A metallic rod of length '$$a$$' is rotated with an angular frequency of $$0.2 \mathrm{~rads}^{-1}$$ about an axis normal to the rod passing through its one end. A constant and uniform magnetic field of '$$\mathrm{B}$$' T parallel to the axis exists everywhere. The emf developed across the ends of the rod is

Accepted Answer

The emf (electromotive force) induced across the ends of a rotating rod in a magnetic field can be calculated using Faraday's law of electromagnetic induction, considering the motion of the rod through the magnetic field.
When the rod rotates about an axis perpendicular to its length with angular frequency $$\omega$$, each element of the rod at a distance $$r$$ from the axis experiences a tangential velocity $$v = r\omega$$. According to the magnetic flux cutting principle, the emf $$d\mathcal{E}$$ induced in a small element of the rod of length $$dr$$ is given by:
$$ d\mathcal{E} = B \cdot v \cdot dr = B \cdot (r\omega) \cdot dr $$
where $$B$$ is the magnetic field.
Integrate this expression from 0 to $$a$$ (the length of the rod) to find the total emf $$\mathcal{E}$$ across the rod:
$$ \mathcal{E} = \int_0^a B \cdot r\omega \cdot dr $$
Perform the integration:
$$ \mathcal{E} = B \omega \int_0^a r \, dr = B \omega \left[ \frac{r^2}{2} ight]_0^a = B \omega \cdot \frac{a^2}{2} $$
Given that $$\omega = 0.2 \ 	ext{rads}^{-1}$$, substitute this value into the equation:
$$ \mathcal{E} = B \cdot 0.2 \cdot \frac{a^2}{2} = \frac{B a^2 \cdot 0.2}{2} $$
Simplify the expression:
$$ \mathcal{E} = \frac{B a^2}{10} $$
Therefore, the emf developed across the ends of the rod is:
Option A:
$$ \frac{\mathrm{Ba}^2}{10} $$

Answer

$$\frac{\mathrm{Ba}^2}{10}$$

Answer

$$\frac{\mathrm{Ba}^2}{5}$$

Answer

$$\frac{\mathrm{Ba}^2}{50}$$

Answer

$$\frac{\mathrm{Ba}^2}{2}$$

Electromagnetic Induction

A metallic rod of length '$a' is rotated with an angular frequency of 0.2 \mathrm{~rads}^{-1} about an axis normal to the rod passing through its one end. A constant and uniform magnetic field of '\mathrm{B}$' T parallel to the axis exists everywhere. The emf developed across the ends of the rod is

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