Physics

The scalar projection of vector B on vector A is given by the dot product of A and B divided by the magnitude of A, which is (A·B)/|A|. This represents the length of the shadow of B onto A.

Heating a magnet causes its atoms to vibrate more, disrupting the alignment of magnetic domains. When heated beyond a certain temperature (Curie temperature), the magnet loses its magnetization completely, effectively demagnetizing it.

Inserting a glass plate (a dielectric) between the plates of a capacitor increases its capacitance, which reduces the overall impedance in the circuit. However, since the capacitor is in series with the bulb, the increased capacitance reduces the current through the bulb, causing its brightness to decrease.

The potential difference across the galvanometer is calculated using Ohm's law: V = IR. Given the resistance R = 100 Ω and the current I = 20 mA = 0.02 A, the potential difference V = 0.02 × 100 = 2 volts. Therefore, the correct answer is option C (2 volt).

The impedance Z of a coil with resistance R and inductance L at frequency f is given by Z = √(R² + (2πfL)²). Option D represents this formula correctly as (R² + 4π²f²L²).

The cross product of two vectors follows the right-hand rule. Pointing your right hand's fingers east and curling them towards south, your thumb points up. Therefore, east × south points in the up direction.

1 Tesla (T) is equal to 10,000 Gauss (G), so the correct relationship is 1T = 10^4 G. This is a standard unit conversion in magnetism.

The expression $\mathbf{i}\cdot (\mathbf{j}\times \mathbf{k})$ equals 1 because $\mathbf{j}\times \mathbf{k}=\mathbf{i}$, and the dot product of $\mathbf{i}$ with itself is 1.

According to Faraday's law of electromagnetic induction, a changing magnetic flux induces an electric field around itself, which can drive currents in conductors. This is the fundamental principle behind electromagnetic induction.

In an AC circuit containing a capacitor, the current leads the voltage by a phase angle of 90 degrees because the capacitor causes the voltage to lag behind the current due to its charging and discharging behavior.