Physics

A null vector is defined as a vector with zero magnitude and no specific direction. Since both zero magnitude and no specified direction are true characteristics, option C is correct.

For three coplanar forces to keep a body in equilibrium, they must be concurrent, meaning their lines of action intersect at a single point. This ensures both the net force and net moment about that point are zero, satisfying equilibrium conditions.

A current-carrying solenoid behaves like a bar magnet with a magnetic field along its axis. When suspended freely, it aligns itself along the Earth's magnetic field, coming to rest in the North-South direction due to magnetic torque.

When a shunt resistance is connected across a galvanometer and the deflection is reduced to half (from 50 divisions to 25 divisions), the galvanometer resistance (Rg) can be calculated using the formula: Rg = (n / (N - n)) * Rs, where N is the original deflection, n is the new deflection, and Rs is the shunt resistance. Here, N = 50, n = 25, and Rs = 12Ω. Substituting these values gives Rg = (25 / (50 - 25)) * 12Ω = (25 / 25) * 12Ω = 12Ω. Therefore, the galvanometer resistance is 12Ω, which corresponds to option D.

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 magnetic field strength inside a solenoid is given by B = μnI, where μ is the permeability, n is the number of turns per unit length, and I is the current. Option A is incorrect because it uses total turns N instead of turns per unit length n, and option B is incorrect due to incorrect formula structure. Therefore, option C is correct.

The vector A = 2i + 2j + 3k has a magnitude greater than 2, calculated as √(2² + 2² + 3²) = √17 ≈ 4.12, so options stating magnitude 2 or -2 are incorrect. However, magnitude cannot be negative, so option D is definitely incorrect. The direction statements need more context but option D is clearly false.

A vector in two-dimensional space can be resolved into two rectangular components along the x and y axes. In three-dimensional space, a vector can be resolved into three mutually perpendicular rectangular components along the x, y, and z axes. Therefore, the maximum number of rectangular components a vector can have in space is 3, not 4. The original answer 'D' (4) is incorrect, and the correct answer is 'C' (3).

The SI unit of magnetic flux density (B) is the tesla (T). One tesla is defined as one weber per square meter (Wb/m²). It can also be expressed in base SI units as newton per ampere per meter (N·A⁻¹·m⁻¹). Among the given options, option B (NAm-1) corresponds to newton per ampere per meter, which is the correct SI unit representation for magnetic flux density. Therefore, the correct answer is B, not D.

The cross product of two vectors $\overrightarrow{a}\times \overrightarrow{b}$ results in a vector perpendicular to both $\overrightarrow{a}$ and $\overrightarrow{b}$, with magnitude $|a||b|\sin \theta$, where $\theta$ is the angle between the vectors measured from $\overrightarrow{a}$ to $\overrightarrow{b}$ using the right-hand rule. The direction follows the right-hand rule, and reversing the order of the vectors reverses the direction of the cross product (i.e., $\overrightarrow{b}\times \overrightarrow{a}=-\overrightarrow{a}\times \overrightarrow{b}$). When the angle between the vectors is between 0° and 180°, $\sin \theta$ is positive, and the cross product points in the positive direction defined by the right-hand rule. However, if the vectors are rotated through 270° (which is equivalent to -90°), the sine of the angle becomes negative, and the cross product vector points in the opposite (negative) direction. The cross product of parallel or anti-parallel vectors is the zero vector, not a negative vector. Therefore, options A and B are incorrect. The cross product of perpendicular vectors has maximum magnitude and points perpendicular to the plane of the vectors but is not negative. Hence, option C is incorrect. The correct answer is D: the cross product is a negative vector when the vectors are rotated through 270°, causing the sine of the angle to be negative and reversing the direction of the cross product vector.

The frequency of an LC circuit is given by f = 1/(2π√(LC)). If the capacitance C is increased four times, the frequency becomes f' = 1/(2π√(L*4C)) = 1/(2π*2√(LC)) = (1/2)f, which is half the original frequency.

The vector A = 2i + 3j has components 2 along x-axis and 3 along y-axis. The angle θ with the y-axis is given by θ = arccos(component along y / magnitude) = arccos(3/√(2²+3²)) = arccos(3/√13) ≈ 60°. Hence, option B is correct.

Antennas typically emit radio waves, which are a type of electromagnetic wave used for communication. While electromagnetic waves is a broader category, the most precise and common answer in this context is radio waves.

Electromagnetic waves travel in free space at the speed of light, which is approximately 3 × 10^8 meters per second. This speed is a universal constant denoted by 'c' and is much greater than the speed of sound waves (approximately 343 m/s in air). Among the given options, 'More than sound waves' (option D) is the correct and most accurate choice, as the other options do not represent speeds or types of electromagnetic waves.

The standard vector symbol for magnetic flux density is $𝐁$. The other options represent different quantities: $𝐇$ is magnetic field strength, $𝐌$ is magnetization, and $𝐋$ is not standard for flux density.

Torque and energy do not have identical dimensions; torque has dimensions of force times distance (ML²T⁻²) but is a vector, while energy is a scalar with the same dimensional formula but different physical meaning. Momentum and impulse share the same dimensions (MLT⁻¹), energy and work are dimensionally identical (ML²T⁻²), and mass and moment of inertia differ since moment of inertia has dimensions of mass times length squared (ML²).

Soft iron is the most suitable material for the core of an electromagnet because it has high magnetic permeability, which allows it to become strongly magnetized and enhances the magnetic field produced by the coil. Air and Cu-Ni alloy have very low magnetic permeability, and steel is harder and retains magnetism, making soft iron preferable for electromagnet cores.

The cross product of the unit vectors $\hat{i}$ and $\hat{j}$ is $\hat{k}$. This is a fundamental property of the right-handed coordinate system in vector algebra.

The original current is I = I₀ sin(πt), where the amplitude is I₀ and angular frequency is π. To double the amplitude and frequency, the amplitude becomes 2I₀ and angular frequency becomes 2π, so the equation is I = 2I₀ sin(2πt).

Turning a steering wheel with both hands is easier because the forces applied by the hands create a couple, producing a torque that helps rotate the wheel more effectively than a single force. This torque (couple) makes it easier to turn the wheel.

The sensitivity of a galvanometer is defined as the deflection per unit current, which is given by the formula S = BAN/C, where B is the magnetic field strength, A is the area of the coil, N is the number of turns, and C is the torsional constant of the suspension wire.

The expression cosθ î + sinθ ĵ represents a vector of unit length pointing at an angle θ from the x-axis, which is the definition of a unit vector in that direction.

In an electromagnetic wave, the direction of propagation is perpendicular to both the electric and magnetic fields, which are themselves perpendicular to each other. This orthogonal relationship is fundamental to the nature of electromagnetic wave propagation.

The maximum current through the galvanometer is calculated using Ohm's law: I = V / R = 100 mV / 50 Ω = 0.1 V / 50 Ω = 0.002 A = 2 mA. Therefore, the correct answer is 2.0 mA.

Using both hands to turn a steering wheel creates a couple, which is a pair of equal and opposite forces whose effect is to produce rotation. This couple generates torque more effectively than a single force, making it easier to turn the wheel.