CUET UG Physics Booster Test – 3 Vector Forces and Superposition

The displacement vector between two points is obtained by subtracting the initial coordinates from the final coordinates. For the x-component: Δx = x₂ − x₁ If movement occurs toward a higher x-value, then x₂ > x₁, making Δx positive. For the y-component: Δy = y₂ − y₁ If movement occurs toward a lower y-value, then y₂ < y₁, making Δy negative. Thus, when moving toward a higher x-value and a lower y-value, the signs of the coordinate differences become: Δx → Positive Δy → Negative This interpretation is important in vector notation because displacement vectors determine both the direction and magnitude of Coulombic interactions. Therefore, Option A is correct.

The unit vectors satisfy: r̂₁₂ = −r̂₂₁ Hence 1 → b. Every unit vector has magnitude 1, therefore: 2 → a. For like charges, the force is repulsive and acts along r̂₂₁: 3 → c. For unlike charges, the force is attractive and acts along −r̂₂₁: 4 → d. These relationships are fundamental to the vector form of Coulomb's law used throughout NCERT. Therefore, the correct matching is: 1-b, 2-a, 3-c, 4-d.

The vector form of Coulomb's law is: F₂₁ = (1/4πε₀)(q₁q₂/r²) r̂₂₁ The sign of q₁q₂ determines whether the interaction is attractive or repulsive. If q₁q₂ is positive, the force is repulsive. If q₁q₂ is negative, the force reverses direction and becomes attractive. Therefore Statements 1 and 2 are correct. Statement 3 is incorrect because a single vector equation handles both cases. Statement 4 is incorrect because the magnitude of a unit vector is always 1 and does not affect the inverse-square dependence. Hence Option A is correct.

According to Newton's third law: F₁₂ = −F₂₁ This means the force exerted by charge q₁ on charge q₂ is exactly equal in magnitude and opposite in direction to the force exerted by q₂ on q₁. Although a proton and an electron have different masses, the forces they exert on each other are equal. Their accelerations differ because acceleration depends on mass, but the force magnitudes remain identical. Therefore Statement B is incorrect because it violates Newton's third law.

Gravitational force always acts attractively between masses. Therefore Statement 1 is correct. Electrostatic force may be attractive or repulsive depending on the signs of the interacting charges, making Statement 2 correct. Both Coulomb's law and Newton's law of gravitation contain a 1/r² dependence, so Statement 3 is also correct. Statement 4 is incorrect because: G = 6.67 × 10⁻¹¹ N m² kg⁻² k ≈ 9 × 10⁹ N m² C⁻² These constants differ enormously in magnitude and units. Hence Statements 1, 2 and 3 are correct.

The ratio (ke²)/(Gmₑmₚ) compares the electrostatic force and gravitational force between an electron and a proton. NCERT shows that this ratio is approximately: 2.4 × 10³⁹ This means the electrostatic interaction is about 10³⁹ times stronger than the corresponding gravitational interaction. Because of this enormous difference, the effect of gravity on the motion of electrons and other charged particles is practically negligible in atomic and microscopic situations. The motion of electrons is controlled almost entirely by electric forces. Therefore, the significance of this ratio is not related to particle masses or the validity of Coulomb's law. It simply demonstrates the overwhelming dominance of electrostatic interactions. Hence Option B is correct.

For a charged particle in an electric field: F = eE and a = F/m = eE/m Since electron mass is much smaller than proton mass, aₑ >> aₚ For motion through the same distance h from rest, h = ½at² or t = √(2h/a) A larger acceleration produces a smaller time of travel. Since electrons have much greater acceleration, they reach the destination sooner. Therefore, the proton takes a longer time because its greater mass reduces its acceleration under the same electric force magnitude. Hence Option A is correct.

Given: aₚ = 1.4 × 10¹⁹ m s⁻² g ≈ 10 m s⁻² Required ratio: aₚ/g = (1.4 × 10¹⁹)/(10) = 1.4 × 10¹⁸ Unit Verification (m s⁻²)/(m s⁻²) = dimensionless This enormous ratio shows that the electrostatic acceleration experienced by a proton is vastly larger than gravitational acceleration. Therefore, gravity has virtually no influence on proton motion in such electric fields. Hence Option C is correct.

The superposition principle states that the force exerted by one charge on another is unaffected by the presence of additional charges. Therefore Statement 1 is correct. Electrostatics contains only pairwise Coulomb interactions. There are no extra three-body or four-body electrostatic forces, making Statement 2 correct. The total force acting on any charge is obtained by vector addition of all pairwise forces. Hence Statement 3 is correct. Statement 4 is incorrect because vacuum permittivity ε₀ is a fundamental constant and does not change with the number of charges present. Thus Statements 1, 2 and 3 are correct.

Electrostatic force possesses both magnitude and direction, making it a vector quantity. Therefore, forces due to different charges must be combined through vector addition. According to the principle of superposition: F = F₁ + F₂ + F₃ + ... This represents a linear vector summation of all individual forces. Electrostatic forces are not multiplied together, nor are they treated as scalar quantities. The vector nature of force is essential because direction plays a crucial role in determining the resultant force. Hence electrostatic force addition is vectorial and follows linear summation. Therefore, Option D is correct.

For θ = 0°, the forces act in the same direction, giving: R = F₁ + F₂ Thus 1 → b. For θ = 180°, the forces act oppositely: R = |F₁ − F₂| Thus 2 → d. For θ = 90°, the resultant becomes: R = √(F₁² + F₂²) Thus 3 → a. For two equal forces acting at an angle θ, symmetry causes the resultant direction to bisect the angle between them. Thus 4 → c. Therefore, the correct matching is: 1-b, 2-d, 3-a, 4-c.

For a system containing n discrete charges, Coulomb's law is applied individually to each pair of interacting charges. The superposition principle is then used to add all the force vectors. For a chosen charge q₁, the total force is: F₁ = F₁₂ + F₁₃ + ... + F₁ₙ This contains exactly n − 1 force terms. Each force possesses a distinct direction represented by a corresponding unit vector such as r̂₁₂, r̂₁₃, etc. Statement 4 is incorrect because integration is required only for continuous charge distributions, not for discrete point-charge systems. Hence Statements 1, 2 and 3 are correct.

The negative charge at C experiences two attractive forces of equal magnitude F from charges at A and B. These forces are separated by an angle of 60° because the triangle is equilateral. Therefore, the resultant force is: R = √(F² + F² + 2F²cos60°) = √(3F²) = √3 F The resultant is not equal to 2F. Statements A, B and D are consistent with the geometry and force directions in the equilateral-triangle configuration. Therefore Statement C is the incorrect statement.

The centroid is equidistant from all three identical charges. Consequently, the test charge experiences three forces of equal magnitude. Because the forces are symmetrically arranged at 120° intervals, their vector sum is zero. Rotating the arrangement about the centroid leaves the physical configuration unchanged. If a non-zero resultant existed, its direction would change under rotation, leading to a contradiction. Therefore the resultant must be zero. Statement 3 is incorrect because each individual force remains non-zero. Only their vector sum vanishes. Thus Statements 1, 2 and 4 are correct.

A symmetric arrangement remains physically unchanged after rotation about its center. The distances between charges and the geometric configuration remain identical. Since the original arrangement produced zero resultant force, the rotated arrangement must also produce zero resultant force. If the force became non-zero after rotation, it would imply that merely changing orientation changes physical laws, which is impossible. The superposition principle remains valid throughout the process, and the individual forces do not change in magnitude. Therefore the resultant force continues to remain zero after rotation. Hence Option A is correct.

In highly symmetric charge configurations, such as charges placed uniformly on a circle or at the vertices of a regular polygon, each force acting on the central charge has a corresponding force that balances it. When all force vectors are added according to the superposition principle, the vector sum becomes zero. A useful geometric interpretation is that if the force vectors are arranged head-to-tail, they form a closed polygon. A closed vector polygon implies that the resultant vector is zero. The cancellation occurs because of symmetry and vector addition, not because the forces themselves vanish. Therefore, the fundamental condition for equilibrium is that the vector sum of all forces must cancel exactly. Hence Option D is correct.

Initially, After the changes: New force: The reduction in charge product by a factor of 4 is exactly compensated by the reduction in distance squared by a factor of 4. Unit Verification Force remains measured in newtons (N). Therefore, the new force remains unchanged. Hence Option B is correct.

When charges are placed inside matter, the atoms and molecules of the medium respond to the electric field. Their charged constituents—electrons and nuclei—interact with the external charges, making the electrostatic situation more complex than in vacuum. Therefore Statements 1 and 3 are correct. Statement 2 is incorrect because the simple vacuum form of Coulomb's law does not automatically account for dielectric effects and polarization. Statement 4 is also incorrect because the superposition principle remains valid in electrostatics. Thus the correct combination contains Statements 1 and 3 only. Hence Option A is correct.

When a charged conductor touches an uncharged conductor, charge transfer occurs through conduction. Electrons move until both conductors reach the same electric potential. For identical conducting spheres, equal potential means equal charge distribution after separation. Initially: Sphere 1 = q Sphere 2 = 0 Total charge = q After contact: Each sphere receives: Thus the process is conduction and the final charges are equal. Hence Option C is correct.

Before contact, the total charge is: Charge conservation requires that the total charge after contact and separation remains: Since the spheres are identical, symmetry requires equal sharing of the total charge. Therefore, each sphere finally possesses: Matching: 1 → c 2 → d 3 → a 4 → b Thus the correct matching is: 1-c, 2-d, 3-a, 4-b.