Electric field of a point charge

Section: Electric Fields | Syllabus: Cambridge A Level Physics (9702)

Field Strength Formula

The electric field E at a distance r from a point charge Q is determined by substituting Coulomb's law into the definition E = F/q.

E = (Q/4π_0 r^2)

The field is radial and follows the inverse square law. For multiple charges, the resultant field is the vector sum of individual fields.

Worked Example: Resultant Field between Spheres

Question: Two spheres (+Q and -Q) are 10.0 cm apart. The resultant field at point X (3.0 cm from +Q) is 6.0 × 10^4 N C^-1. Determine Q.

Solution

1. Both fields point to the right (away from +Q, towards -Q).

2. E_total = E_+ + E_- = (Q/4π_0 r_1^2) + (Q/4π_0 r_2^2).

3. 6.0 × 10^4 = (Q/4π_0) ( (1/0.03^2) + (1/0.07^2) ).

4. 6.0 × 10^4 = Q × (8.99 × 10^9) × (1111.1 + 204.1).

5. Q = 6.0 × 10^41.18 × 10^13 = 5.1 × 10^-9 C.

Worked Example: Perpendicular Fields

Question: Point P is 40 mm from two +5.0 nC charges in perpendicular directions. Determine the strength of the resultant field.

Solution

1. E_1 = E_2 = (kQ/r^2) = 8.99 × 10^9 × 5.0 × 10^-90.04^2 = 2.81 × 10^4 N C^-1.

2. Resultant field E = √(E_1^2 + E_2^2) = E_1 √(2).

3. E = 2.81 × 10^4 × 1.41 = 3.97 × 10^4 N C^-1.

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