Coulomb's Law (force between multiple charges)

Coulomb force is a vector, so both magnitude and direction must be included. First select the target charge and use Coulomb’s law separately for each other charge, F = k|qQ|/r². Resolve the forces into components when needed and add the components algebraically. Option A is correct; simply adding magnitudes ignores direction, selecting only the largest charge omits valid contributions, and mutual external-charge forces do not act directly on the target.

Coulomb’s law gives F = kq₁q₂/r². Let the original force be F. Multiplying one charge by 8 multiplies the force by 8, while multiplying the separation by 6 divides it by 6² = 36. Thus F′/F = 8/36 = 2/9. Therefore F′ = 216 × 2/9 = 48 N. Option B is correct; 24 N would correspond to a factor 1/9, while the larger choices ignore the inverse-square effect sufficiently.

Using F = kq₁q₂/r², multiplying each charge by 4 multiplies the product q₁q₂ by 4 × 4 = 16. Increasing distance eightfold divides the force by 8² = 64. Hence F′/F = 16/64 = 1/4, so F′ = 160/4 = 40 N. Option C is correct. The other values result from treating charge or distance changes linearly instead of using the charge product and inverse-square dependence.

Coulomb’s law can be written F ∝ (q₁q₂)/r². For the force to remain constant when the charge product becomes 81 times larger, r′² must also become 81 times larger. Taking the positive physical length ratio gives r′/r = √81 = 9. Therefore the separation must become nine times its original value, so option A is correct. An 81-fold distance change would reduce the force too much.

From Coulomb’s law, F′/F = (q₁′q₂′/q₁q₂)(r/r′)². Here the charge-product factor is 64 and the required force factor is 4, so 4 = 64(r/r′)². Thus (r′/r)² = 16 and r′/r = 4. The distance must therefore become four times, making option A correct. Eight times would make the force only one-fourth of the original, not four times.

For two equal forces F making an angle θ, the resultant is R = √(F² + F² + 2F²cosθ). At θ = 60°, cosθ = 1/2, so R = √(3F²) = F√3. Given R = 40√3 N, division by √3 gives F = 40 N. Thus option B is correct; 40√3 N is the resultant itself, not each component.

Perpendicular forces combine by the Pythagorean theorem. If each force has magnitude F, then R = √(F² + F²) = F√2. Substituting R = 35√2 N gives F√2 = 35√2, hence F = 35 N. Option B is correct. The value 35√2 N is the resultant, while 17.5 N and 70 N do not satisfy the required vector relation.

The selected positive charge is repelled by the other two charges, so two force vectors of 60 N act along the two sides meeting at its corner. The angle between these vectors is 60°. Therefore R = √(60² + 60² + 2·60²cos60°) = 60√3 N. Option C is correct; the forces do not cancel because their directions are separated by 60°, not 180°.

The two specified forces are equal and perpendicular, so their resultant is found using the Pythagorean theorem: R = √(25² + 25²) = √(2×625) = 25√2 N. Therefore option C is correct. Adding the magnitudes directly would incorrectly give 50 N, while zero would require equal opposite directions rather than perpendicular directions.

Both source charges are positive, so they repel the positive target in opposite directions. The left charge pushes the target right with a proportional factor 32/4² = 32/16 = 2. The right charge pushes it left with factor 8/2² = 8/4 = 2. The magnitudes are equal and directions opposite, so the vector sum is zero. Option A is correct.

The governing concept is superposition: the net force is calculated on one specified target charge. First identify that charge and define a sign or coordinate direction. Then calculate the individual Coulomb forces produced by every other charge and add them vectorially. The colours or mutual forces between external charges are irrelevant to the requested force, while choosing only the farthest charge omits necessary interactions.

The governing principle is Newton’s second law applied to the selected target charge: its net electric force is the vector sum of only the forces exerted on that charge by all other charges. A mutual force between two external charges acts on those external charges, not on the target. Therefore option A is correct. Such external mutual forces may affect the motion of the source charges, but they do not belong in the target’s force balance. Options B, C and D give incorrect reasons.