physics : class12 Superposition of Dipole Fields : Superposition Principle : Electrostatics MCQ Questions and Answers
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Superposition of Dipole Fields
MCQ questions and answers.
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MCQ Questions
1. Two identical point dipoles, each having dipole moment p in the +x direction, are located at (0, +a, 0) and (0, -a, 0). What is the electric field at the origin? Let k = 1/(4πε0).
A.
−2kp/a^3 in the +x direction
B.
2kp/a^3 in the +x direction
C.
−2kp/a^3 in the −x direction
D.
Zero
Correct Answer:
−2kp/a^3 in the +x direction
Explanation:
The origin lies on the equatorial line of both dipoles. The field due to each dipole is −kp/a^3 in the x direction because E = k[3(p·r̂)r̂ − p]/r^3 and p·r̂ = 0. Adding the two fields gives −2kp/a^3 in the +x direction, equivalently 2kp/a^3 in the −x direction.
2. A dipole p in the +x direction is at the origin, and another dipole of moment 2p in the +x direction is at x = a. What is the electric field at x = 2a on the common axis? Let k = 1/(4πε0).
A.
15kp/(4a^3) in the +x direction
B.
17kp/(4a^3) in the +x direction
C.
9kp/(4a^3) in the +x direction
D.
7kp/(4a^3) in the +x direction
Correct Answer:
17kp/(4a^3) in the +x direction
Explanation:
The field on the axial side of a dipole is 2kp/r^3 along the dipole moment. The first dipole contributes 2kp/(2a)^3 = kp/(4a^3), while the second contributes 2k(2p)/a^3 = 4kp/a^3. Their sum is 17kp/(4a^3) in the +x direction.
3. Two point dipoles are placed at the same point. Their moments are p in the +x direction and 2p in the +y direction. What is the field at a point on the +z axis at distance r? Let k = 1/(4πε0).
A.
kp/r^3 (x̂ + 2ŷ)
B.
−kp/r^3 (x̂ + 2ŷ)
C.
kp/r^3 (2x̂ − ŷ)
D.
−kp/r^3 (x̂ − 2ŷ)
Correct Answer:
−kp/r^3 (x̂ + 2ŷ)
Explanation:
For each dipole, the observation point lies on its equatorial line because the displacement is along z while both dipole moments lie in the xy-plane. Thus each field is −k p_vector/r^3. The resultant is −k(p x̂ + 2p ŷ)/r^3.
4. Two identical dipoles, each with moment p in the +x direction, are located at (0, +a, 0) and (0, −a, 0). What is the magnitude and direction of the electric field at the origin? Let k = 1/(4πε0).
A.
2kp/a^3, in the −x direction
B.
2kp/a^3, in the +x direction
C.
4kp/a^3, in the −x direction
D.
kp/a^3, in the −x direction
Correct Answer:
2kp/a^3, in the −x direction
Explanation:
Each dipole produces an equatorial field of magnitude kp/a^3 opposite to its dipole moment. Both contributions therefore point in the −x direction and add, giving 2kp/a^3.
5. A dipole p in the +x direction is at the origin, while an oppositely oriented dipole −p is at x = a. At the midpoint between them, what is the resultant electric field? Let k = 1/(4πε0).
A.
Zero
B.
16kp/a^3 in the +x direction
C.
16kp/a^3 in the −x direction
D.
8kp/a^3 in the +x direction
Correct Answer:
Zero
Explanation:
At the midpoint, each dipole is at distance a/2. The first dipole produces an axial field 2kp/(a/2)^3 = 16kp/a^3 in the +x direction. The second dipole has moment −p, so its axial field at the point to its left is equal and opposite, giving zero resultant field.
6. Two dipoles are located at the origin with moments p x̂ and p ŷ. At a point on the +y axis a distance r from the origin, what is the resultant field? Let k = 1/(4πε0).
A.
kp/r^3 (x̂ + 2ŷ)
B.
kp/r^3 (−x̂ + 2ŷ)
C.
kp/r^3 (x̂ − 2ŷ)
D.
−kp/r^3 (x̂ + 2ŷ)
Correct Answer:
kp/r^3 (−x̂ + 2ŷ)
Explanation:
For the dipole p x̂, the point lies on its equatorial line, producing −kp/r^3 x̂. For the dipole p ŷ, the point lies on its axial line, producing 2kp/r^3 ŷ. Superposition gives kp/r^3(−x̂ + 2ŷ).
7. Three point dipoles are placed at the origin with moments p x̂, p ŷ, and −p ẑ. What is the electric field at a point on the +z axis at distance r? Let k = 1/(4πε0).
A.
kp/r^3 (−x̂ − ŷ + 2ẑ)
B.
kp/r^3 (−x̂ − ŷ − 2ẑ)
C.
kp/r^3 (x̂ + ŷ + 2ẑ)
D.
kp/r^3 (−x̂ + ŷ + 2ẑ)
Correct Answer:
kp/r^3 (−x̂ − ŷ − 2ẑ)
Explanation:
The x̂ and ŷ dipoles produce equatorial fields −kp x̂/r^3 and −kp ŷ/r^3. The dipole −p ẑ is antiparallel to the observation direction, so its axial field is −2kp ẑ/r^3. Adding all three gives the stated vector.
8. Two identical dipoles of moment p in the +x direction are located at x = −a and x = +a. What is the electric field at the origin? Let k = 1/(4πε0).
A.
Zero
B.
4kp/a^3 in the +x direction
C.
2kp/a^3 in the +x direction
D.
4kp/a^3 in the −x direction
Correct Answer:
4kp/a^3 in the +x direction
Explanation:
The origin is on the axial line of both dipoles. For the dipole at x = −a, the field at the origin is 2kp/a^3 in the +x direction. For the dipole at x = +a, the point is on the opposite axial side, but the axial field still points along the dipole moment, also contributing 2kp/a^3 in +x. The total is 4kp/a^3.
9. A dipole p x̂ is at the origin and a dipole −2p x̂ is at x = a. What is the electric field at x = 2a? Let k = 1/(4πε0).
A.
−15kp/(4a^3) in the +x direction
B.
−15kp/(4a^3) in the −x direction
C.
17kp/(4a^3) in the +x direction
D.
−7kp/(4a^3) in the +x direction
Correct Answer:
−15kp/(4a^3) in the +x direction
Explanation:
The first dipole contributes kp/(4a^3) in +x because its distance is 2a. The second dipole has moment −2p and is observed on its +x axial side, so it contributes −4kp/a^3. The sum is kp/(4a^3) − 4kp/a^3 = −15kp/(4a^3), directed along −x.
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