CUET UG Physics Booster Test – 2 Vector Forces and Superposition

The position vectors are: r₁ = 2i + 3j r₂ = 5i + 7j The displacement vector from q₁ to q₂ is: r₂₁ = r₂ − r₁ = (5 − 2)i + (7 − 3)j = 3i + 4j Magnitude: |r₂₁| = √(3² + 4²) = √(9 + 16) = √25 = 5 m This displacement vector gives the separation between the two charges and is used extensively in the vector form of Coulomb's law. NCERT uses such vector notation to determine both distance and direction between interacting point charges. Therefore, the magnitude of r₂₁ is 5 m.

The unit vector r̂₁₂ is directed from charge 1 toward charge 2, while r̂₂₁ points from charge 2 toward charge 1. Therefore: r̂₂₁ = −r̂₁₂ Unit vectors are used in Coulomb's law to specify the direction of electrostatic force. Thus Statements 2 and 3 are correct. Statement 4 is incorrect because for unlike charges the force is attractive and acts opposite to the chosen displacement unit vector. Therefore, the force is not always strictly along r̂₂₁. According to the option structure provided, the correct answer is Option D.

For two like charges, the electrostatic force is repulsive. Therefore, the force on charge 2 due to charge 1 acts away from charge 1 and along the unit vector r̂₂₁. For two unlike charges, the force is attractive. Hence the force on charge 2 acts toward charge 1 and therefore along −r̂₂₁. The vector form of Coulomb's law automatically accounts for attraction and repulsion through the sign of the product q₁q₂. Thus: Like charges → Along r̂₂₁ Unlike charges → Along −r̂₂₁ Therefore Option C is correct.

Electrostatic interactions satisfy Newton's third law, giving: F₁₂ = −F₂₁ Thus 1 → b. For an isolated system, the vector sum of all internal forces is zero due to action-reaction pairs, giving 2 → d. The superposition principle together with Coulomb's law is used to calculate forces in multi-charge systems, so 3 → c. Repulsive force occurs when both charges have the same sign, giving 4 → a. Therefore the correct matching is: 1-b, 2-d, 3-c, 4-a.

Both Coulomb's law and Newton's law of gravitation have an inverse-square dependence on distance. Hence Statement 1 is correct. The ratio of electrostatic force to gravitational force for elementary particles is extremely large, showing that electrostatic interactions are enormously stronger than gravitational interactions. Therefore Statement 3 is correct. Gravity acts only attractively between masses, making Statement 4 correct. Statement 2 is incorrect because the electrostatic force between two protons is vastly stronger than the gravitational force. Thus Statements 1, 3 and 4 are correct.

For an electron and a proton: Fe = ke²/r² Fg = Gmₑmₚ/r² Therefore, Fe/Fg = ke²/Gmₑmₚ Notice that the distance r² cancels completely. Hence the ratio is independent of separation distance and depends only on fundamental constants. NCERT gives: Fe/Fg ≈ 2.4 × 10³⁹ This enormous value demonstrates that electrostatic interactions dominate microscopic motion, making gravitational effects negligible for charged particles such as electrons. Therefore, Statement B is incorrect because the ratio does not require the exact distance between particles.

For a charged particle in a uniform electric field: F = eE a = eE/m Using: s = ½at² t = √(2s/a) Substituting acceleration: t = √(2sm/eE) Thus time depends on particle mass and is inversely proportional to √E. Because a proton is much heavier than an electron, it experiences a smaller acceleration and therefore takes a longer time to travel the same distance. This differs from gravitational free fall where all bodies experience the same acceleration g (ignoring air resistance), making fall time independent of mass. Hence all four statements are correct.

The electrostatic force on a proton is: F = eE Using: e = 1.6 × 10⁻¹⁹ C E = 2.0 × 10⁴ N C⁻¹ F = 3.2 × 10⁻¹⁵ N Acceleration: a = F/mₚ = (3.2 × 10⁻¹⁵)/(1.67 × 10⁻²⁷) ≈ 1.9 × 10¹² m s⁻² This value is enormously larger than g = 9.8 m s⁻². Therefore gravitational effects are negligible when compared with electrostatic effects in microscopic systems. 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, the force between q₁ and q₂ remains exactly the same whether a third charge q₃ is present or not. When extra charges are introduced, only the resultant force changes because new force vectors are added. The original pairwise force remains unchanged. This principle greatly simplifies electrostatic calculations because each interaction can be evaluated independently before performing vector addition. Hence Option B correctly represents the superposition principle.

When two forces act along the same line but in opposite directions, the resultant magnitude is obtained by subtraction. Given: F₁ = 3.6 × 10⁴ N F₂ = 4 × 10³ N Resultant: F = F₁ − F₂ = 3.6 × 10⁴ − 0.4 × 10⁴ = 3.2 × 10⁴ N The direction of the resultant force is the same as the direction of the larger force. This is a direct application of vector addition and the superposition principle discussed in NCERT. Therefore, the magnitude of the resultant force is 3.2 × 10⁴ N.

According to NCERT, the force on a charge due to multiple charges is obtained using the principle of superposition. Each individual Coulomb force is first calculated separately and then added vectorially. The parallelogram law is one of the standard methods of vector addition. It automatically incorporates both the magnitudes of the forces and the angle between them. Thus Statements 1 and 2 are correct. Statement 3 is incorrect because the parallelogram law is particularly useful when vectors are not collinear. Statement 4 is incorrect because the superposition principle remains the fundamental principle; the parallelogram law is only a mathematical tool used for vector addition. Hence Option A is correct.

Consider a system containing n point charges. If we wish to calculate the net force acting on one selected charge, that charge experiences forces due to every other charge in the system. Since one charge has already been chosen as the target charge, the remaining charges available to exert forces are: n − 1 Thus the resultant force is: F = F₁₂ + F₁₃ + ... + F₁ₙ which contains exactly n − 1 force terms. The total number of charges in the system remains n, while the number of contributing force vectors acting on the chosen charge is n − 1. Therefore Option A is correct.

For an equilateral triangle of side l: Altitude: AD = (√3/2) l Hence 1 → c. The centroid divides the median in the ratio 2:1, giving the distance from a vertex to the centroid as: AO = l/√3 Hence 2 → d. When equal repulsive forces act symmetrically on a vertex charge, the resultant lies along the angle bisector. Thus 3 → a. The individual forces are added using vector addition methods such as the parallelogram law. Therefore 4 → b. Hence the correct matching is: 1-c, 2-d, 3-a, 4-b.

The centroid of an equilateral triangle is equidistant from all three vertices. Therefore, if equal charges are placed at the vertices, the charge Q at the centroid experiences three forces of equal magnitude. These forces are separated by angles of 120°. Their vector sum is exactly zero because of symmetry. Rotating the entire system does not change any physical property of the arrangement. Hence the zero-force condition remains unchanged. Therefore Statement 3 is incorrect. Statements 1, 2 and 4 are correct and together explain why the resultant force on the centroid charge vanishes. Hence Option A is correct.

An isolated system of charges obeys Newton's third law. Every force exerted by one charge on another has an equal and opposite reaction force. As a result, the vector sum of all internal forces in the complete system is zero. This conclusion remains true regardless of how the system is oriented in space. Rotating the entire charge arrangement by 60° does not change any relative distances, angles or physical interactions. Therefore, the total force cannot suddenly become non-zero merely because of rotation. Hence Statement C is incorrect and is the correct answer.

For a charge placed at a center of symmetry, forces due to surrounding charges are arranged symmetrically. According to the superposition principle, the resultant force equals the vector sum of all individual forces. If the resultant force were non-zero, rotating the entire symmetric configuration would change the force direction without changing the physical arrangement, which is impossible. Hence Statement 1 is correct. Symmetry ensures that the individual force vectors cancel when added vectorially, making Statement 2 correct. Statement 4 is also correct because the result follows directly from applying the superposition principle to a symmetric geometry. Statement 3 is incorrect because electric field lines do not physically break or disappear. Therefore, Statements 1, 2 and 4 are correct.

According to Coulomb's law, the electrostatic force between unlike charges is attractive. The unit vector r̂₂₁ points from charge q₂ toward charge q₁. Since attraction pulls the charge toward the other charge, the force acts opposite to the chosen direction of the separation vector. Therefore: F₂₁ ∝ −r̂₂₁ The force always acts along the line joining the two charges and never perpendicular to it. Hence Option B correctly describes the direction of the force for unlike charges.

NCERT explains that when charges are placed in matter instead of vacuum, the charged constituents of the medium such as electrons and nuclei influence the electrostatic interaction. The medium may become polarized and modify the electric field produced by the charges. As a result, the electrostatic force between charges in matter becomes more complicated than in free space. This is why dielectric properties are introduced in later discussions of electrostatics. Therefore, Option D is correct because the presence of matter affects the interaction through its internal charged particles.

Initially: Sphere A = q Sphere B = 0 After A and B are touched, the total charge q is shared equally because the spheres are identical. Therefore: A = q/2 B = q/2 Now Sphere B carrying q/2 is touched to identical uncharged Sphere C. Total charge involved: q/2 + 0 = q/2 Since B and C are identical, this charge is equally divided. Charge on each sphere after separation: (q/2)/2 = q/4 Therefore, the final charge remaining on Sphere B is q/4. This is a direct application of charge conservation and equal charge sharing among identical conducting spheres.

When two identical conducting spheres are brought into contact, charges move until both spheres reach the same electric potential. Since the spheres are identical in size and shape, symmetry requires that they finally possess equal charges. The total charge before and after contact remains unchanged because of the law of conservation of charge. Therefore, equal sharing results from two principles: 1. Geometrical symmetry of identical spheres. 2. Conservation of total charge. Statements 3 and 4 are unrelated to the reason for equal charge sharing in this situation. Hence Statements 1 and 2 are correct.