An electric field is a vector field surrounding a charged object. It tells us the magnitude and direction of the electrostatic force that a charge would experience at each point.
The charge that produces the field is called the source charge. A second charge placed in the field to investigate it is called the test charge.
Source charge and test charge
Charge
Role
Source charge Q
Produces the electric field in the surrounding region
Test charge q
Experiences force and is used to determine the field at a point
By convention, the test charge is positive. Its direction of force therefore defines the direction of E⃗. It must be extremely small so that its presence does not significantly redistribute charge or alter the original source field.
Definition of electric field intensity
Electric field intensity at a point is the electrostatic force experienced per unit positive test charge placed at that point.
Key equation
E=Fq
\vec E
electric field intensity at the point (N·C⁻¹)
\vec F
electrostatic force on the test charge (N)
q
positive test-charge magnitude (C)
When to use: Use when the force on a known positive test charge is given.
Key equation
F=qE
When to use: Use to find the force on a charge placed in a known field.
For a negative charge placed in the field, q is negative in F⃗=qE⃗, so its force points opposite to the field direction.
Field due to a point source charge
For a source charge Q and a small positive test charge q separated by distance r, Coulomb's law gives the force:
Key equation
F=kQqr2r
\hat r
radial unit vector directed away from the source (none)
Dividing by q gives the electric field of the source charge. The test-charge magnitude cancels.
Key equation
E=kQr2r=14πε0Qr2r
Q
source charge (C)
r
distance from the source charge (m)
k
Coulomb constant (N·m²·C⁻²)
Key equation
E=k|Q|r2
When to use: Use this form for field magnitude; determine direction separately from the sign of Q.
Direction of electric field
Source charge
Direction of E⃗
Q>0
Radially outward, because a positive test charge is repelled
Q<0
Radially inward, because a positive test charge is attracted
More generally, electric field direction is described from positive charge toward negative charge. At every point, it is the direction in which a unit positive test charge would move or tend to move.
Units and dependence
Key equation
[E]=NC=N·C-1=V·m-1
Change
Effect on field magnitude
Q→2Q at fixed r
E→2E
r→2r at fixed Q
E→E/4
r→3r at fixed Q
E→E/9
r becomes very large
E approaches zero and becomes practically negligible
The field of a point charge extends through space, although its strength decreases rapidly with distance according to the inverse-square law.
Textbook Example 10.2
Determine the electric field at the surface of a sphere of radius 3.0 m when a point charge of 9 μC is placed at its centre.
Key equation
Q=9μC=9×10-6C,\qquad r=3.0m
Key equation
E=kQr2=9×1099×10-6(3.0)2
Key equation
E=9×103N·C-1
Because the central charge is positive, the field at the surface is directed radially outward.
Assignment 10.2 as guided practice
A conducting sphere carries a charge of 200 μC. Find the electric field intensity at a distance of 60 cm from its centre.
Key equation
Q=200μC=200×10-6C,\qquad r=60cm=0.60m
Key equation
E=9×109200×10-6(0.60)2=5.0×106N·C-1
Exam-ready method
Identify the source charge Q and observation distance r.
Convert charge to coulombs and distance to metres.
Use E=k|Q|/r² for the magnitude.
State outward direction for Q>0 or inward direction for Q<0.
If force and test charge are given, use E⃗=F⃗/q.
Write the final unit as N·C⁻¹ or V·m⁻¹.
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Exam preparation
Exam Focus
Important points
Definitions of electric field and field intensity.
Small positive test-charge convention.
E⃗=F⃗/q and E⃗=kQ r̂/r².
Units N·C⁻¹ and V·m⁻¹.
Outward/inward direction and Example 10.2.
Common mistakes
Including the test charge q in the final field formula E=k|Q|/r².
Defining field direction using a negative test charge instead of a positive one.
Writing inward field for a positive source or outward field for a negative source.
Forgetting to square the distance from the source charge.
Using centimetres or microcoulombs without converting to SI units.
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.
Electric Field Intensity
Sticky Notes — Quick Revision
1.
Key Point 1
An electric field is the region around a source charge in which another charge experiences electrostatic force.
2.
Key Point 2
Electric field intensity is force per unit positive test charge: E⃗=F⃗/q.
3.
Key Point 3
For a point charge, E⃗=kQ r̂/r².
4.
Key Point 4
The field points away from a positive charge and toward a negative charge.
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8 concept questions
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Explanation after each answer
Exam Practice
Important Question
Q1. Define electric field and electric field intensity.
Derivation
Q2. Derive the expression for electric field intensity due to a point charge.
Concept
Q3. Why is a small positive test charge used to define electric field direction?
Numerical
Q4. Solve textbook Example 10.2.
Reasoning
Q5. How does electric field intensity depend on source charge and distance?
