Physics

In three-dimensional space, the direction of a vector is specified by two angles, often called direction cosines or angles with respect to coordinate axes. Three angles would be redundant, and one angle or no angle are insufficient to specify direction in space.

In a capacitive circuit, the current leads the voltage by 90 degrees (π/2 radians) because the capacitor causes the current to change before the voltage does. This is a fundamental property of capacitors in AC circuits.

The self-inductance L of a coil is given by the formula L = emf / (rate of change of current). Given an induced emf of 2 V and a rate of change of current of 4 A/s, the self-inductance is L = 2 V / 4 A/s = 0.5 H. However, the options given include 0.5 mH (0.0005 H) and 0.5 H. The correct calculation yields 0.5 H, which corresponds to option B, not A. The original explanation had a mismatch between the calculation and the option letter. Therefore, the correct answer is B (0.5 H).

Given vectors A = 3i + 2j - k and B = 3i - 2j + k, the dot product is calculated as (3)(3) + (2)(-2) + (-1)(1) = 9 - 4 - 1 = 4, which is not zero. Therefore, vectors A and B are not perpendicular, so option D is incorrect. B is not a negative vector of A (which would mean B = -A), so option B is incorrect. B does not have a negative magnitude since magnitude is always positive, so option C is incorrect. However, B is exactly the negative of A, i.e., B = -A, which means B is anti-parallel to A. Hence, option A is correct.

A galvanometer is an instrument used to measure the current flowing through a circuit, not just detect its presence. It provides a quantitative measurement of current, typically small currents, by deflection of a needle.

When a charged particle moves perpendicular to a uniform magnetic field, it experiences a centripetal force equal to the magnetic Lorentz force qvB. Equating mv^2/R = qvB and solving for R gives R = mv/(qB).

A Wheatstone bridge is balanced when there is no potential difference across the galvanometer branch, resulting in zero current flow through it. This condition allows accurate measurement of unknown resistance.

A central force acts along the line joining the centers of two bodies and therefore cannot produce torque about the center of force, as torque requires a force component perpendicular to the radius vector.

A galvanometer is designed to be very sensitive to small currents, which is achieved by having very low damping. Low damping allows the coil to deflect easily in response to current without quickly losing energy to resistive forces.

In a pure resistor circuit, the voltage and current are in phase, meaning they have zero phase difference. This means the voltage and current reach their maximum and zero values simultaneously.