Physics

An ammeter connected in an AC circuit measures the RMS (Root Mean Square) value of the current because RMS value represents the effective current that produces the same heating effect as a DC current of the same value.

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.

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.

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

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 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.

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 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).

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.

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.

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.

A magnetic field exerts a force on a charged particle only when the particle is moving and its velocity has a component perpendicular to the magnetic field lines. If the particle moves parallel to the field lines, no magnetic force acts on it. Therefore, option D is correct.

A unit vector denoted as n^ typically represents a normal vector to a surface, meaning it is perpendicular to that surface. This is a standard notation in physics and vector mathematics.

The maximum number of components a vector can have depends on the dimension of the vector space it belongs to. In an n-dimensional vector space, a vector has exactly n components. However, vector spaces can be infinite-dimensional, meaning vectors can have infinitely many components. Therefore, the maximum number of components a vector can have is infinite, not limited to 3 as in three-dimensional space.

Vector addition obeys the commutative law (order doesn't matter), associative law (grouping doesn't matter), and distributive law (scalar multiplication distributes over vector addition). Therefore, all given laws apply to vector addition.

The dot product of two vectors A and B is calculated as the sum of the products of their corresponding components: AxBx + AyBy + AzBz. Option A correctly represents this formula.

The magnitude of the vector $𝐀=2\hat{i}-2\hat{j}+\hat{k}$ is calculated as $\sqrt{2^2+(-2{)}^2+1^2}=\sqrt{4+4+1}=\sqrt{9}=3$. The unit vector in the direction of $𝐀$ is the vector divided by its magnitude, which is $\frac{2}{3}\hat{i}-\frac{2}{3}\hat{j}+\frac{1}{3}\hat{k}$. This corresponds to option C, $\frac{2\hat{i}-2\hat{j}+\hat{k}}{3}$. Therefore, the correct answer is C.

An alternating current (AC) not only changes its direction repeatedly but also varies its magnitude in a sinusoidal (harmonic) manner over time. Option D accurately describes both these characteristics, making it the most complete and correct answer.

The cross product of two vectors results in a vector perpendicular to both, with direction given by the right-hand rule. The cross product is anti-commutative, meaning $𝐚\times 𝐛=-𝐛\times 𝐚$. This implies that reversing the order of the vectors reverses the direction (sign) of the cross product vector. Rotating one vector through 270° relative to the other is equivalent to rotating it -90°, which reverses the direction of the cross product vector, effectively producing a 'negative' vector. When vectors are parallel or anti-parallel, the sine of the angle between them is zero, so the cross product is the zero vector, not a negative vector. When vectors are perpendicular, the cross product vector has maximum magnitude and a definite direction. Therefore, the cross product is a negative vector when the vectors are rotated through 270°, i.e., when the order or orientation causes the direction to reverse.

When the coil of a galvanometer is in equilibrium, the deflecting couple caused by the current is exactly balanced by the restoring couple produced by the spring or suspension. This balance of torques results in no net rotation, maintaining equilibrium.

A vector can be multiplied by a scalar, which can be dimensionless or dimensional, and can also be negative. Therefore, all options A, B, and C are correct.

For two equal forces F acting at an angle θ, the resultant is R = √(F² + F² + 2F²cosθ) = F√(2 + 2cosθ). Given R = F, so √(2 + 2cosθ) = 1. Squaring gives 2 + 2cosθ = 1, hence cosθ = -1/2. Therefore, θ = 120°.

For a sinusoidal alternating current, the peak current (I_peak) is related to the RMS current (I_rms) by the formula I_peak = I_rms × √2. Given I_rms = 4.0 A, the peak current is I_peak = 4.0 × 1.414 = 5.66 A approximately. The resistor value and frequency do not affect this calculation. Therefore, the correct answer is D (5.66 A).

An electromagnetic wave consists of electric and magnetic fields oscillating perpendicular to each other and to the direction of wave propagation. This perpendicular arrangement is fundamental to the nature of electromagnetic waves.

An electron moving through perpendicular electric and magnetic fields experiences forces from both fields. It will suffer no deflection if the electric force equals the magnetic force in magnitude but opposite in direction, which gives E = Bv (where v is the electron's velocity).