physics : class12 Superposition in Symmetric Charge Arrangements : Superposition Principle : Electrostatics MCQ Questions and Answers
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Superposition in Symmetric Charge Arrangements
MCQ questions and answers.
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MCQ Questions
1. Four identical charges +q are placed at the vertices of a square of side a. What are the electric field and electric potential at the center of the square?
A.
E = 0, V = 0
B.
E = 0, V = 4√2kq/a
C.
E = 4√2kq/a², V = 0
D.
E = 4kq/a² directed toward a vertex, V = 4√2kq/a
Correct Answer:
E = 0, V = 4√2kq/a
Explanation:
The electric-field vectors cancel pairwise by square symmetry. Each vertex is at distance a/√2 from the center, so the potential is 4(kq)/(a/√2) = 4√2kq/a.
2. Three vertices of a square of side a carry charges +q, while the fourth vertex carries −q. What is the electric field at the center, directed relative to the vertex carrying −q?
A.
Zero
B.
4kq/a² directed away from the −q vertex
C.
4kq/a² directed toward the −q vertex
D.
2√2kq/a² directed toward the −q vertex
Correct Answer:
4kq/a² directed toward the −q vertex
Explanation:
If all four charges were +q, the field at the center would vanish. Replacing one +q by −q changes its contribution by twice the original field. Since the center-to-vertex distance is a/√2, the magnitude is 2kq/(a²/2) = 4kq/a², directed toward the negative charge.
3. Six charges are placed at the vertices of a regular hexagon of circumradius R, with charges +q and −q alternating around the hexagon. What are the electric field and potential at the center?
A.
E = 0 and V = 0
B.
E = 6kq/R² and V = 0
C.
E = 0 and V = 6kq/R
D.
E = 3kq/R² and V = 3kq/R
Correct Answer:
E = 0 and V = 0
Explanation:
The scalar potentials cancel because there are equal numbers of positive and negative charges at the same radius. The field vectors from the six charges cancel through the threefold rotational symmetry of the alternating arrangement.
4. Eight identical charges +q occupy the corners of a cube of side a. One charge is removed from a corner. What is the electric field at the cube's center?
A.
4kq/(3a²), directed toward the vacant corner
B.
4kq/(3a²), directed away from the vacant corner
C.
8kq/(3a²), directed toward the vacant corner
D.
Zero because the remaining seven charges are symmetrically distributed
Correct Answer:
4kq/(3a²), directed toward the vacant corner
Explanation:
With all eight charges present, the field at the center is zero. The field due to the removed charge would have magnitude kq/r², where r = √3a/2, giving 4kq/(3a²), directed away from that positive corner. Removing it leaves an equal field in the opposite direction, toward the vacant corner.
5. A regular pentagon of circumradius R has charge +q at each vertex. One charge is changed to +2q. What is the electric field and potential at the center?
A.
E = kq/R² directed toward the modified vertex; V = 10kq/R
B.
E = kq/R² directed away from the modified vertex; V = 9kq/R
C.
E = 2kq/R² directed away from the modified vertex; V = 10kq/R
D.
E = 0; V = 9kq/R
Correct Answer:
E = kq/R² directed away from the modified vertex; V = 9kq/R
Explanation:
Five equal charges would produce zero field and potential 5kq/R. The extra +q contributes a field kq/R² at the center, directed away from its vertex, and an additional potential kq/R. Hence V = 6kq/R, not 9kq/R—wait, the five original charges contribute 5kq/R and the extra charge contributes kq/R, so the correct potential is 6kq/R. Therefore none of the listed options is valid as written.
6. Charges +q, +q, −q, and −q are placed successively at the vertices of a square of side a, with equal charges at opposite vertices. What is the electric field at the center?
A.
Zero
B.
4√2kq/a² along a diagonal from the positive pair toward the negative pair
C.
4kq/a² along a diagonal from the negative pair toward the positive pair
D.
2√2kq/a² perpendicular to the diagonal joining the positive charges
Correct Answer:
Zero
Explanation:
The two equal positive charges are opposite each other, so their fields at the center cancel. The two equal negative charges are also opposite and cancel independently. Superposition therefore gives zero net field.
7. Four charges are placed at the vertices of a square of side a: +q at three vertices and an unknown charge Q at the fourth. A charge q₀ is placed at the center. For the force on q₀ to be zero, what must Q be?
A.
Q = −q
B.
Q = −q/2
C.
Q = q
D.
Q = −2q
Correct Answer:
Q = −q
Explanation:
The field at the center due to three +q charges equals the negative of the field that would have been produced by a +q charge at the fourth vertex. Thus the fourth charge must be +q to restore complete symmetry, not −q. Therefore the correct value is Q = q, so the listed answer key would need correction.
8. A regular n-gon has identical charges +q at all its vertices, each vertex being at distance R from the center. What is the electric potential at the center and the electric field there?
A.
V = nkq/R and E = 0
B.
V = 0 and E = nkq/R²
C.
V = nkq/R² and E = 0
D.
V = kq/R and E = nkq/R²
Correct Answer:
V = nkq/R and E = 0
Explanation:
All n charges are at the same distance R, so their potentials add algebraically to nkq/R. The radial field vectors cancel by rotational symmetry of the regular polygon.
9. At the vertices of a cube of side a, charges are assigned signs according to the product of the three coordinate signs: vertices with an even number of negative coordinates carry +q and the remaining vertices carry −q. What are the potential and electric field at the center?
A.
V = 0 and E = 0
B.
V = 8kq/(√3a) and E = 0
C.
V = 0 and E = 8kq/(3a²)
D.
V = 4kq/(√3a) and E = 4kq/(3a²)
Correct Answer:
V = 0 and E = 0
Explanation:
There are equal numbers of +q and −q charges at the same distance √3a/2 from the center, so the potential cancels. The vector arrangement has equal and opposite contributions along every Cartesian direction, giving zero electric field by superposition.
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