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Topic 2 of 15

Electric Fields

Rather than thinking of one charge exerting a direct "action at a distance" force on another, it is far more powerful to introduce the electric field: a property of space itself. A source charge creates a field everywhere around it; a second charge then responds to the field at its location. This field concept becomes indispensable when we get to time-varying fields and electromagnetic waves.

Key Concepts

Electric Field Definition

The electric field

\vec{E}

at a point is defined as the force per unit positive test charge placed at that point:

\vec{E} = \vec{F}/q_0

. Units: N/C (equivalently V/m). The test charge

q_0

must be small enough not to disturb the source charges.

Field of a Point Charge

A point charge

q

creates a field

E = k|q|/r^2

at distance

r

. The field points radially outward from a positive charge and inward toward a negative charge.

Electric Field Lines

Field lines are a visual representation: they point in the direction of

\vec{E}

, originate on positive charges and terminate on negative charges, never cross, and are denser where the field is stronger.

Superposition of Fields

The total electric field at a point due to several charges is the vector sum of the individual fields:

\vec{E}_{\text{net}} = \vec{E}_1 + \vec{E}_2 + \cdots

. For continuous distributions, this sum becomes an integral.

Electric Dipole

Two equal and opposite charges

+q

and

-q

separated by distance

d

. The dipole moment is

\vec{p} = q\vec{d}

(pointing from

-q

+q

), with magnitude

p = qd

. Dipoles in a uniform external field experience a torque

\tau = pE\sin\theta

that tends to align them with the field.

Key Equations

Electric field from force

\vec{E} = \frac{\vec{F}}{q_0}

Definition of the electric field at a point in terms of force on a positive test charge q₀.

Field of a point charge

E = k\frac{|q|}{r^2}

Magnitude of the field at distance r from point charge q. Direction: away from positive q, toward negative q.

Electric dipole moment

p = qd

Magnitude of the dipole moment. p points from −q to +q. Torque in field: τ = pE sin θ.

Torque on dipole in uniform field

\tau = pE\sin\theta

Torque on a dipole of moment p in field E; θ is the angle between p and E. Tends to align the dipole with the field.

Worked Example

Net Electric Field Between Two Opposite Charges

Problem

Charge $q_1 = +5$ nC is at the origin and $q_2 = -5$ nC is at $x = 0.20$ m. Find the electric field at the midpoint $x = 0.10$ m.

Solution

Field from $q_1$ (positive, at origin) points away from it — in the $+x$ direction at $x = 0.10$ m:

E_1 = k\frac{q_1}{r^2} = (8.99\times10^9)\frac{5\times10^{-9}}{(0.10)^2} = 4495 \text{ N/C}

Field from $q_2$ (negative, at $x = 0.20$ m) points toward it — also in the $+x$ direction at the midpoint:

E_2 = k\frac{|q_2|}{r^2} = (8.99\times10^9)\frac{5\times10^{-9}}{(0.10)^2} = 4495 \text{ N/C}

Both fields point in the same direction (+x), so they add:

E_{\text{net}} = E_1 + E_2 = 4495 + 4495 = 8990 \text{ N/C}

Answer E_net = 8990 N/C in the +x direction.

Practice

Exercises

7 problems

1 of 7

A point charge $q = +8$ nC. What is the magnitude of the electric field (in N/C) at a distance $r = 0.40$ m from it?

Your answer

N/C

2 of 7

A charge $q_0 = +3\,\mu\text{C}$ is placed in a uniform electric field $E = 2.0\times10^5$ N/C. What is the magnitude of the force (in N) on it?

Your answer

3 of 7

Charges $q_1 = +5$ nC (at $x = 0$ ) and $q_2 = -5$ nC (at $x = 0.20$ m) form a dipole. What is the magnitude of the electric field (in N/C) at the midpoint $x = 0.10$ m?

Your answer

N/C

4 of 7

Charges $q_1 = +4$ nC (at $x = 0$ ) and $q_2 = +1$ nC (at $x = 0.60$ m) are fixed. At what position $x$ (in m) between them is the electric field zero?

Your answer

5 of 7

A dipole consists of $q = +5$ nC and $-5$ nC separated by $d = 0.040$ m. What is the dipole moment (in nC·m)?

Your answer

nC·m

6 of 7

A charge $q = -4\,\mu\text{C}$ is placed in a uniform field $E = 600$ N/C. What is the magnitude of the force (in mN) on the charge?

Your answer

7 of 7

The electric field at a distance of $r = 0.15$ m from a point charge is measured to be $E = 800$ N/C. What is the magnitude of the charge (in nC)?

Your answer

Key Takeaways

The electric field is a property of space — it exists at every point whether or not a test charge is placed there.
At any point, $\vec{F} = q\vec{E}$ : positive charges are pushed along $\vec{E}$ ; negative charges are pushed opposite to $\vec{E}$ .
For opposite-sign charge pairs (dipoles), the fields at the midpoint add; for same-sign pairs, they partially cancel.
Field lines can never cross — if they did, a test charge at that point would have to accelerate in two directions simultaneously.
To find a null-field point between two like charges, set the two field magnitudes equal and solve for position.