Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R):
Assertion (A): An electron in a certain region of uniform magnetic field is moving with constant velocity in a straight line path.
Reason (R): The magnetic field in that region is along the direction of velocity of the electron.
In the light of the above statements, choose the correct answer from the options given below:
- Assertion (A) is true because an electron moving in a straight line with constant velocity in the presence of a magnetic field must not experience any force in the direction of motion. This implies the velocity of the electron is perpendicular to the magnetic field, so there is no magnetic force component along the velocity.
- Reason (R) is also true since the magnetic field must be perpendicular to the velocity for the force to not affect the motion of the electron. However, the statement that the magnetic field is "along the direction of velocity" contradicts the nature of the magnetic force, which acts perpendicular to both the magnetic field and the velocity. Thus, Reason (R) does not correctly explain Assertion (A).
Final Answer: Both (A) and (R) are true, but (R) is not the correct explanation of (A).
A current element X is connected across an AC source of emf \(V = V_0\ sin\ 2πνt\). It is found that the voltage leads the current in phase by \(\frac{π}{ 2}\) radian. If element X was replaced by element Y, the voltage lags behind the current in phase by \(\frac{π}{ 2}\) radian.
(I) Identify elements X and Y by drawing phasor diagrams.
(II) Obtain the condition of resonance when both elements X and Y are connected in series to the source and obtain expression for resonant frequency. What is the impedance value in this case?
Let $ P_n = \alpha^n + \beta^n $, $ n \in \mathbb{N} $. If $ P_{10} = 123,\ P_9 = 76,\ P_8 = 47 $ and $ P_1 = 1 $, then the quadratic equation having roots $ \alpha $ and $ \frac{1}{\beta} $ is: