Coulomb's Law (force between multiple charges)

Choose east as the positive x-direction and north as the positive y-direction. The force components are Fx = +15 N and Fy = +20 N. Both components are positive, so their vector sum lies in the first quadrant, between east and north; therefore it points north-east. Its magnitude would be √(15² + 20²) = 25 N, while the angle above east is tan⁻¹(20/15), but neither calculation changes the quadrant.

The positive source charge above repels the positive target, pushing it downward. The negative source charge below attracts the positive target, also pulling it downward. Because the source charges have equal magnitudes and equal distances, the two force magnitudes are equal, but their directions are the same rather than opposite. They therefore add vertically downward. The forces do not cancel; cancellation would require opposite force directions.

The negative charge above repels the negative target, so it pushes the target downward. The positive charge below attracts the negative target, also pulling it downward toward the lower charge. Equal magnitudes and equal distances make the two forces equal in size, but both point downward, so they reinforce one another. Consequently, the net force is downward, not zero and not upward.

The centre is equidistant from the two opposite corners. Since the source charges are equal, Coulomb’s law gives equal force magnitudes on the target charge. The two forces act along the same diagonal but in opposite directions: one source pushes or pulls along the diagonal one way, and the opposite source produces the corresponding force the other way. Their vector sum is therefore zero, regardless of the target charge’s sign.

The governing idea is Coulomb’s law combined with superposition. Equal original charges at equal distances produce equal opposite repulsive forces on the positive middle charge, so they initially cancel. After the left charge is tripled, its repulsion is three times the original leftward or rightward contribution: it pushes the middle charge to the right more strongly than the unchanged right charge pushes it left. Hence option B is correct.

By Coulomb’s law, opposite charges attract. Thus the left positive charge pulls the negative middle charge leftward, while the right positive charge pulls it rightward. Initially these forces would be equal because the distances and charges are equal. Doubling the right charge doubles the rightward attraction, whereas the leftward attraction is unchanged. The rightward force is therefore larger, so option B gives the net direction.

The governing concept is vector addition of perpendicular forces. The eastward and northward forces are at 90°, so the resultant magnitude follows the Pythagorean relation: R = √(9² + 12²) = √(81 + 144) = √225 = 15 N. Adding 9 and 12 directly would give 21 N, but that is not valid for perpendicular vectors. Therefore option B is correct.

Vector addition determines the direction of the resultant. The horizontal component points west and the vertical component points south, so their sum must lie in the quadrant between west and south. Therefore it points southwest. The magnitudes affect the exact angle, with tan θ = 8/15 measured below the west direction, but they cannot move the resultant into another quadrant. Hence option B is correct.

For perpendicular force components, the resultant magnitude is obtained from the Pythagorean theorem. Thus R = √(24² + 32²) = √(576 + 1024) = √1600 = 40 N. The difference, 8 N, and the direct sum, 56 N, are not the correct magnitudes for perpendicular vectors; 768 N is a multiplication-related distractor. Therefore option B is correct.

The direction of the vector sum of two equal forces is determined by symmetry. Each force contributes the same magnitude, so the resultant must make equal angles with both original directions. The only such direction inside the angle is its angle bisector; for a 45° separation, the resultant is 22.5° from either force. It is not along one force or opposite to both. Therefore option C is correct.

The governing concept is vector addition of forces. The opposite eastward and westward forces first give a net horizontal component of 20 − 12 = 8 N eastward. This 8 N component is perpendicular to the 15 N northward component. Therefore, the resultant magnitude is √(8² + 15²) = √289 = 17 N. Hence option A is correct; simply adding all magnitudes would ignore direction.

The governing idea is vector addition with attention to direction. The northward and southward forces oppose each other, leaving 13 − 5 = 8 N northward. This remaining component is perpendicular to the 15 N eastward force. Thus the net magnitude is √(8² + 15²) = √289 = 17 N. Option A is correct; 23 N would result from an inappropriate scalar addition.

For two equal forces F making an angle θ, the resultant magnitude is R = √(F² + F² + 2F² cos θ) = 2F cos(θ/2). Complete cancellation requires θ = 180°, so the forces must be exactly opposite. Since the stated angle is less than 180°, R is nonzero. Therefore option B is correct; units and the word charge do not determine cancellation.

Coulomb’s law states F = k|q₁q₂|/r². With both charges fixed, F is proportional to 1/r². If F′ = 36F, then (r/r′)² = 36, so r′ = r/6. The separation must therefore be reduced to one-sixth of its original value, making option B correct. Increasing the distance would decrease, not increase, the force.

By Coulomb’s law, F ∝ 1/r² when the charges do not change. To obtain F′ = F/25, the squared distance must increase by 25: (r′/r)² = 25. Taking the positive physical distance gives r′/r = 5, so the separation must become five times larger. Therefore option A is correct; reducing the distance would increase the force.

Coulomb’s law gives F = kq₁q₂/r². Increasing one charge by 16 multiplies the numerator by 16. To keep F unchanged, r² must also become 16 times larger. Hence r′ = √16 r = 4r. The distance must be four times the original distance, so option A is correct; leaving distance unchanged would make the force 16 times larger.

According to Coulomb’s law, F ∝ q₁q₂/r². Reducing each charge to one-third makes the charge product one-ninth of its original value. To restore the force, r² must also become one-ninth, which means r′ = r/3. Thus the distance must become one-third, making option B correct. Keeping the distance unchanged would reduce the force to one-ninth.

The governing principle is vector addition. Let the first two forces be +F and −F along the same line. Their vector sum is exactly zero because they have equal magnitudes and opposite directions. Adding the third, perpendicular force G gives R = 0 + G = G. Therefore the net force equals the third force, so option A is correct; the perpendicular relationship does not change the cancellation of the first pair.

For perpendicular force components, the resultant follows the Pythagorean theorem: R = √(F₁² + F₂²). Substituting the values gives R = √(12² + 35²) = √(144 + 1225) = √1369 = 37 N. Thus option C is correct. Adding 12 and 35 would incorrectly treat perpendicular vectors as parallel, while multiplying them is not a resultant-force rule.

The governing concept is perpendicular vector addition. The resultant magnitude is R = √(21² + 28²) = √(441 + 784) = √1225 = 35 N. Therefore option B is correct. The difference, 7 N, would apply to opposite collinear forces, and the sum, 49 N, would apply only to parallel forces in the same direction. Multiplication is not appropriate here.

The governing concept is vector equilibrium: the vector sum of all forces must be zero. Since the first two forces are perpendicular, their resultant has magnitude R = √(6² + 8²) = √100 = 10 N. The third force must be equal in magnitude and opposite in direction to this resultant, so its magnitude is 10 N. The other choices come from subtraction, addition, or multiplication rather than perpendicular vector addition.

Coulomb’s law gives F = kq₁q₂/r². The distance changes from 2 to 4 units, so it doubles and the force would become one-fourth if the charges were unchanged. To restore the original force, the product q₁q₂ must increase fourfold. Because the charges are equal, multiplying each by 2 makes their product 2 × 2 = 4 times larger. Therefore option A is correct; multiplying each by 4 would make the product sixteen times larger.

Coulomb’s law states F = kq²/r² for two equal charges. If the separation becomes r/2, the factor 1/r² makes the force four times larger when q is unchanged. To compensate, q² must become one-fourth of its original value. Thus each equal charge must become half, because (q/2)² = q²/4. Option B would reduce the product too much, while keeping the charges unchanged would not preserve the force.

A resultant is found by adding force components as vectors. The horizontal component is positive toward the east and the vertical component is positive toward the north. Therefore the resultant must lie in the quadrant between east and north, namely the north-east direction. Its magnitude would be √(7² + 24²) = 25 N, although only direction is requested. North-west and southward options contain a component with the wrong sign, so option A is correct.

The governing principle is superposition: the net electrostatic force on a target charge equals the vector sum of the forces produced by all other charges. Select the target first, apply Coulomb’s law to each source charge, and record both magnitude and direction. Only then add components or vectors. Option A is correct; B ignores direction, C omits valid forces, and D includes forces that do not act on the target.