Questions: Superposition Principle in Electrostatics

Question 1

Two equal positive charges are placed symmetrically on the x-axis: one at x = +d and one at x = −d. What is the net electric field at the origin due to both charges?

• A. Zero — the fields from the two charges point in opposite directions and cancel exactly
• B. Double the field from a single charge, in the +x direction
• C. Double the field from a single charge, directed outward from the origin in both directions simultaneously
• D. The field cannot be determined without knowing the exact charge magnitude

Answer: (A)

Each positive charge creates a field that points away from it. At the origin, the charge at +d pushes a positive test charge in the −x direction; the charge at −d pushes a positive test charge in the +x direction. These two contributions are equal in magnitude and exactly opposite in direction, so they cancel to zero by vector addition. This is the key operation in superposition: add the individual vector contributions. The magnitudes being equal does not mean the result is 2|E| — direction matters.

Question 2

Three point charges produce fields of magnitudes |E₁| = 5 N/C, |E₂| = 5 N/C, and |E₃| = 5 N/C at point P. The net electric field at P is...

• A. Not necessarily 15 N/C — the net field is the vector sum and depends on the directions of E₁, E₂, and E₃
• B. Exactly 15 N/C — fields from independent sources always add their magnitudes
• C. 5 N/C — equal contributions from independent sources average out by symmetry
• D. 0 N/C — three charges in symmetric arrangement always produce zero net field

Answer: (A)

Superposition requires adding fields as vectors, not scalars. If all three fields point in the same direction, the result is 15 N/C. If they point 120° apart (like three equally spaced charges on a circle), the result is 0 N/C. Any intermediate geometry gives something in between. The magnitudes alone are insufficient — direction determines the result. This is the most common error when applying superposition: treating electric fields as numbers rather than arrows.

Question 3

The presence of a second charge does not alter the electric force that a first charge exerts on a test charge.

• T. True
• F. False

Answer: True

True — this is the empirical content of the superposition principle. Each pair of charges interacts independently of all others. Adding a second source charge to the scene does not modify the force the first charge exerts. The total force on the test charge is the vector sum of the individual forces, each of which is identical to what it would be in isolation. This independence is an experimentally verified fact about electrostatics, not a logical necessity.

Question 4

When two electric field vectors at a point have equal magnitudes but point in opposite directions, the net field magnitude equals twice the magnitude of either individual field.

• T. True
• F. False

Answer: False

False. Two vectors with equal magnitudes and exactly opposite directions sum to zero — they cancel completely. The net field magnitude is 0, not 2|E|. The net magnitude equals 2|E| only when both vectors point in exactly the same direction. This underscores why direction is essential: identical magnitudes can combine to anything from 0 to 2|E| depending on the angle between them.

Question 5

A student adds the magnitudes of two electric field vectors to find the net field. What error has the student made, and under what special condition would this approach give the correct answer?

Model answer: The student treated electric fields as scalars rather than vectors. Electric field is a vector quantity with both magnitude and direction; superposition requires adding the vector components, not the magnitudes. Scalar addition of magnitudes overestimates the net field in any case where the fields are not perfectly aligned. The approach gives the correct answer only when both field vectors point in exactly the same direction (angle = 0° between them), because then |E₁ + E₂| = |E₁| + |E₂|.

The correct procedure is: (1) decompose each field vector into x and y components, (2) sum all x-components to get E_net,x and sum all y-components to get E_net,y, (3) compute the magnitude of the resultant as √(E_net,x² + E_net,y²). This component method works for any number of fields in any directions and is the standard approach for superposition problems.