AP Physics C: Electricity and Magnetism
Electric Field
Learning Objectives
By the end of this lesson, students should be able to:
-
Define electric field and electric field strength.
-
Calculate the electric field produced by a point charge.
-
Determine the direction of electric fields.
-
Apply the principle of superposition to electric fields.
-
Analyze electric field diagrams and field lines.
-
Solve AP Physics C problems involving electric fields.
Introduction to Electric Fields
What is an Electric Field?
An electric field is a region of space surrounding a charged object in which another charge experiences an electric force.
Instead of thinking about charges acting on each other across empty space, physicists describe the interaction using electric fields.
A charged object creates an electric field throughout the surrounding space.
When another charge enters that field, it experiences an electric force.
Why Electric Fields Are Useful
Electric fields allow us to describe the effect of a charge at every point in space.
The electric field exists whether or not another charge is present.
This concept is fundamental to understanding:
-
Electric forces
-
Electric potential
-
Capacitors
-
Electromagnetic waves
Definition of Electric Field
Electric Field Strength
Electric field strength is defined as the electric force per unit positive test charge.
Mathematically:
$$
E=\frac{F}{q}
$$
where:
-
\(E\) = electric field (N/C)
-
\(F\) = electric force (N)
-
\(q\) = test charge (C)
The SI unit of electric field is:
$$
N/C
$$
or equivalently:
$$
V/m
$$
Test Charge
A test charge is a small positive charge used to measure an electric field.
The test charge must be small enough that it does not significantly alter the existing field.
By convention:
-
Electric field direction is the direction a positive test charge would move.
Electric Field Produced by a Point Charge
Electric Field Equation
A point charge produces an electric field whose magnitude is:
$$
E=k\frac{|Q|}{r^2}
$$
where:
-
\(E\) = electric field strength
-
\(Q\) = source charge
-
\(r\) = distance from the charge
-
\(k=8.99\times10^9;N\cdot m^2/C^2\)
Inverse-Square Relationship
Because distance is squared:
$$E\propto\frac{1}{r^2}$$
This means:
-
Doubling the distance reduces the field to one-fourth.
-
Tripling the distance reduces the field to one-ninth.
-
Halving the distance increases the field by four times.
Direction of Electric Fields
Positive Source Charge
For a positive charge:
Electric field lines point away from the charge.
A positive test charge would be repelled.
Field direction:
Outward
Negative Source Charge
For a negative charge:
Electric field lines point toward the charge.
A positive test charge would be attracted.
Field direction:
Inward
Summary of Directions
Positive charge:
-
Field points away from the charge.
Negative charge:
-
Field points toward the charge.
Always remember:
Electric field direction is defined using a positive test charge.
Electric Field Lines
Characteristics of Field Lines
Electric field lines help visualize electric fields.
Properties:
-
Begin on positive charges.
-
End on negative charges.
-
Never cross.
-
Show field direction.
-
Become denser where the field is stronger.
Field Strength and Line Density
The closer the field lines are together, the stronger the electric field.
The farther apart the field lines are, the weaker the electric field.
Field line density is proportional to field strength.
Relationship Between Electric Force and Electric Field
Force on a Charge in an Electric Field
If a charge is placed in an electric field:
$$
F=qE
$$
where:
-
\(F\) = electric force
-
\(q\) = charge
-
\(E\) = electric field
Positive Charges
For a positive charge:
The force points in the same direction as the electric field.
Negative Charges
For a negative charge:
The force points opposite the electric field.
This distinction is commonly tested on AP Physics exams.
Principle of Superposition for Electric Fields
Multiple Charges
When multiple charges are present, the total electric field is the vector sum of the individual fields.
$$
\vec{E}_{net}=\sum \vec{E}
$$
This is known as the Principle of Superposition.
Steps for Solving Superposition Problems
-
Calculate the electric field produced by each charge.
-
Determine the direction of each field.
-
Resolve into components if necessary.
-
Add the vectors.
Example 1: Electric Field from a Point Charge
Problem
A charge of
$$
Q=+5.0\times10^{-6}C
$$
is located at the origin.
Find the electric field 0.20 m away.
Solution
Use:
$$
E=k\frac{|Q|}{r^2}
$$
Substitute values:
$$
E=(8.99\times10^9)\frac{5.0\times10^{-6}}{(0.20)^2}
$$
$$
E=1.12\times10^6N/C
$$
Answer
$$
E=1.12\times10^6N/C
$$
Direction:
Away from the positive charge.
Example 2: Force in an Electric Field
Problem
A charge
$$
q=2.0\times10^{-6}C
$$
is placed in an electric field of
$$
E=3.0\times10^4N/C
$$
Find the force.
Solution
Use:
$$
F=qE
$$
Substitute values:
$$
F=(2.0\times10^{-6})(3.0\times10^4)
$$
$$
F=0.060N
$$
Answer
$$
F=0.060N
$$
Because the charge is positive, the force points in the same direction as the field.
Electric Field Between Parallel Plates
Uniform Electric Field
Between two large parallel plates:
-
The electric field is approximately uniform.
-
Field lines are parallel.
-
Field strength remains nearly constant.
The electric field is:
$$
E=\frac{\Delta V}{d}
$$
where:
-
\(\Delta V\) = potential difference
-
\(d\) = plate separation
This equation becomes important when studying capacitors.
Common AP Exam Mistakes
Mistake 1
Confusing electric force and electric field.
Remember:
$$
F=qE
$$
Force depends on the test charge.
Electric field does not.
Mistake 2
Forgetting direction.
Always determine whether:
-
The source charge is positive or negative.
-
The test charge is positive or negative.
Mistake 3
Ignoring vector addition.
Electric fields must be added as vectors.
Do not simply add magnitudes unless all fields point in the same direction.
AP Free-Response Strategy
Draw a Diagram
Always sketch:
-
Charges
-
Distances
-
Field directions
A diagram often reveals the correct solution path.
Label Directions Carefully
For every electric field:
-
Identify the source charge.
-
Determine whether the field points toward or away from the source.
Check Units
Common units:
$$
N/C
$$
$$
V/m
$$
Both units represent electric field strength.
Summary
Key Takeaways
-
Electric field is defined as force per unit charge.
-
Electric field strength is given by:
$$
E=\frac{F}{q}
$$
-
A point charge produces an electric field:
$$
E=k\frac{|Q|}{r^2}
$$
-
Positive charges create outward fields.
-
Negative charges create inward fields.
-
Electric field lines visualize field strength and direction.
-
The force on a charge in an electric field is:
$$
F=qE
$$
-
Multiple electric fields combine according to the Principle of Superposition.
-
Electric fields are fundamental to understanding all later topics in AP Physics C: Electricity and Magnetism.