3bElectric Fields
Learn how electric fields describe the influence of charges in space, how to calculate field strength, and how to use superposition to find the net field from multiple charges.
The concept of the electric field, introduced by Michael Faraday, revolutionized physics by replacing 'action at a distance' with local field interactions. Electric fields are used in cathode ray tubes, particle accelerators, inkjet printers, and the ion channels of every nerve cell in your body.
How can one charged object exert a force on another across empty space — and what is the "field" that carries this interaction?
Lesson Overview
The electric field is a vector field that describes the force per unit positive charge at every point in space. It is defined as E = F/q₀, where q₀ is a small positive test charge. The SI unit is N/C (or equivalently V/m). The electric field due to a point charge Q at distance r is E = kQ/r², directed away from positive charges and toward negative charges. Electric field lines are visual representations: they point in the direction of E, their density indicates field strength, and they never cross. For a uniform electric field (between parallel plates), E is constant and field lines are parallel. The superposition principle states that the total electric field at a point is the vector sum of fields from all individual charges.
Key Equations
Worked Examples
A +3.0 μC charge experiences a force of 0.12 N to the right when placed at a point P. Find the electric field at P.
Find the electric field at a point 0.30 m from a +5.0 μC point charge.
Two charges: q₁ = +4.0 μC at x = 0 and q₂ = −4.0 μC at x = 0.40 m. Find the electric field at the midpoint (x = 0.20 m).
An electron (charge −e = −1.6×10⁻¹⁹ C) is placed in a uniform electric field of 2.0×10⁴ N/C pointing to the right. Find the force on the electron.
Describe the electric field lines for (a) an isolated positive charge, (b) an isolated negative charge, and (c) two equal and opposite charges (electric dipole).
Guided Problems
At what distance from a +2.0 μC charge is the electric field equal to 8.0×10⁴ N/C?
Hint: Use E = kQ/r² → r² = kQ/E. Solve for r.
Two charges q₁ = +3.0 μC at x = 0 and q₂ = +3.0 μC at x = 0.60 m. Find the electric field at x = 0.30 m (the midpoint).
Hint: Both charges produce fields at the midpoint. E₁ points in +x (away from q₁); E₂ points in −x (away from q₂). They are equal and opposite — what is the net field?
A proton is placed in a uniform electric field E = 5.0×10⁴ N/C. Find its acceleration. (m_p = 1.67×10⁻²⁷ kg)
Hint: F = qE = eE. Then use Newton's second law: a = F/m.
Sketch the electric field lines for two positive charges of equal magnitude placed 10 cm apart. Describe the field at the midpoint.
Hint: Field lines leave both positive charges. At the midpoint, the fields from each charge point in opposite directions and cancel (E = 0 at the midpoint for equal charges).
A uniform electric field of 3.0×10³ N/C points in the +x direction. A charge of −2.0 μC is placed in this field. Find the force on the charge and describe its motion if released from rest.
Hint: F = qE. Since q is negative, F is in the −x direction (opposite to E). The charge accelerates in the −x direction.
Key Vocabulary
Electric Field
A vector field that represents the force per unit positive charge at each point in space. E = F/q₀. Units: N/C or V/m. The field exists whether or not a test charge is present.
Example: Near a +1 μC charge at 0.10 m, E = kQ/r² = 9×10⁵ N/C, pointing away from the charge.
Test Charge
A hypothetical small positive charge used to probe the electric field at a point. It must be small enough not to disturb the field it is measuring.
Example: To measure the field at a point, we imagine placing a tiny +1 nC test charge there and measuring the force on it.
Electric Field Lines
Imaginary lines whose direction at every point is the direction of the electric field. They start on positive charges and end on negative charges. Closer spacing = stronger field.
Example: Between the plates of a capacitor, field lines are parallel and equally spaced, indicating a uniform electric field.
Uniform Electric Field
An electric field that has the same magnitude and direction at every point. Produced between two large parallel plates with equal and opposite charges. E = V/d.
Example: Inside a parallel-plate capacitor with voltage 100 V and plate separation 5 mm, E = 100/0.005 = 20,000 N/C, uniform and perpendicular to the plates.
Superposition Principle
The total electric field at a point due to multiple charges is the vector sum of the fields produced by each charge individually.
Example: At a point equidistant from a +q and −q charge, the fields from each charge both point in the same direction (from + to −), so they add.
Electric Dipole
A pair of equal and opposite charges separated by a small distance. Characterized by its dipole moment p = qd, pointing from negative to positive charge.
Example: A water molecule is a permanent electric dipole — the oxygen end is slightly negative and the hydrogen end is slightly positive.
Workbook Check — Interactive Quiz
Interactive Practice — 5 Questions
The electric field at a point is defined as:
The electric field 0.20 m from a +4.0 μC charge is:
Electric field lines always:
A −3.0 μC charge is placed in a field of 2.0×10⁴ N/C pointing right. The force on the charge is:
At the midpoint between two equal positive charges, the electric field is:
Independent Practice
Find the electric field at a point 0.50 m from a −8.0 μC charge. State the magnitude and direction.
A charge of +2.0 μC is placed in a uniform electric field of 5.0×10⁴ N/C. Find the force on the charge. If the charge is released from rest, find its acceleration (m = 0.010 kg).
Two charges: q₁ = +6.0 μC at x = 0 and q₂ = −6.0 μC at x = 0.60 m. Find the electric field at (a) x = 0.30 m and (b) x = −0.30 m.
Explain why electric field lines cannot cross each other. What would it mean physically if they did?
★ Three charges are at the corners of an equilateral triangle with side 0.40 m: q₁ = +2.0 μC at (0,0), q₂ = +2.0 μC at (0.40,0), q₃ = −4.0 μC at (0.20, 0.346). Find the electric field at the centroid of the triangle (0.20, 0.115 m).
ChallengeCommon Mistakes
Confusing electric field (E) with electric force (F)
E = F/q is the field (property of space); F = qE is the force on a specific charge. The field exists independently of any test charge.
Assuming the electric field direction is always away from the source charge
E points away from positive source charges and toward negative source charges. Always identify the sign of the source charge first.
Adding electric field magnitudes without considering direction
Electric fields are vectors. Use superposition with vector addition — fields in opposite directions partially or fully cancel.
Thinking a larger test charge gives a larger electric field reading
E = F/q₀ is independent of q₀. A larger test charge experiences a larger force, but E = F/q₀ remains the same. The field is a property of the source, not the test charge.
Math Tips
E = F/q (definition) and E = kQ/r² (point charge). Both give N/C. Use the first to find E from a measured force; use the second to calculate E from a source charge.
For superposition: find E from each charge separately (magnitude and direction), then add as vectors. Draw a diagram to get directions right.
F = qE: if q is positive, F is in the same direction as E. If q is negative, F is opposite to E.
At the midpoint between two equal charges: if both are positive (or both negative), E = 0 by symmetry. If they are opposite, the fields add.