Electric Charges and Fields — Questions & Answers

Sample questions from Electric Charges and Fields

1. Q1. Two dipoles of equal moment p are placed at right angles to each other at the same location. The magnitude of the resultant dipole moment is:

   • A.2p
   • B.Zero
   • C.p
   • D.sqrt(2)*p✓
   Solution

   Dipole moment is a vector and moments superpose by vector addition. Two perpendicular vectors of equal magnitude p give a resultant sqrt(p^2 + p^2) = sqrt(2)*p, bisecting the right angle.

2. Q2. The same dipole (p = 8 x 10^-11 C.m) at the same distance (30 cm) but on the EQUATORIAL line produces a field of magnitude closest to:

   • A.53 N/C
   • B.27 N/C✓
   • C.80 N/C
   • D.106.7 N/C
   Solution

   E_equatorial = kp/r^3 = (9x10^9)(8x10^-11)/(0.3)^3 = 26.7 N/C approx 26.7 N/C -- exactly half the axial field at the same distance, which is a key relationship for short dipoles (E_axial = 2 x E_equatorial at the same r). A student who forgets the factor of 2 relationship and just guesses the axial value again picks A.

3. Q3. An electric dipole is placed in a NON-uniform electric field, with its dipole moment vector not aligned with the field direction. Compared to a dipole in a UNIFORM field under the same conditions, this dipole will experience:

   • A.Only a torque, no net force, same as in a uniform field
   • B.Only a net force, no torque
   • C.Both a net force and a torque✓
   • D.Neither a force nor a torque
   Solution

   In a uniform field, a dipole not aligned with the field experiences a torque but ZERO net translational force (since the equal-and-opposite forces on +q and -q are also equal in magnitude, as the field magnitude is the same at both charge locations). In a NON-uniform field, however, the field magnitude differs slightly at the +q and -q locations (since they're at slightly different points in a field that varies with position), so the two oppositely-directed forces on the charges do NOT exactly cancel, resulting in a net force on the dipole IN ADDITION TO the torque. A student who assumes dipoles always behave the same regardless of field uniformity picks A.

4. Q4. A solid non-conducting sphere of radius R has a spherical cavity of radius R/2 carved out, the cavity centre lying at distance R/2 from the sphere centre. The field everywhere inside the cavity is:

   • A.Uniform, of magnitude rho*R/(6*eps0)✓
   • B.Zero
   • C.Radial and proportional to distance from the cavity centre
   • D.Radial and proportional to 1/r^2
   Solution

   Superpose a full sphere of density rho and a sphere of density -rho filling the cavity. Inside a uniform sphere E = rho*r/(3*eps0) measured from its own centre, so the two contributions give a constant rho*a/(3*eps0) where a = R/2 is the centre separation. That is rho*R/(6*eps0), uniform in magnitude and direction throughout the cavity.

5. Q5. Continuous charge distribution has three types. Surface charge density σ is:

   • A.C/m
   • B.C/m²✓
   • C.C/m³
   • D.C/s
   Solution

   σ = charge per unit area (C/m²). λ = C/m (linear). ρ = C/m³ (volume).

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