EM-03 · Electrostatics field

The Electric Field

A charge creates a vector field, the electric field, in the space around it. You confirm that each arrow is the force a unit positive charge would feel at that spot, and that this force weakens with the square of the distance.

Change the distance and the sign, and read the field

The ball in the middle is the source charge, and the arrows around it are the electric field it creates. Moving the test charge closer or farther with the slider changes the length of the gold force arrow, and pressing a sign button flips the direction of every arrow at once.

The buttons change the sign of the source charge.

Distance r to the test charger = 1.60

Drag to rotate; the slider sets the distance and the buttons set the sign of the charge.

The field E at this point

Outward (away from the charge)

E = kQ/r² r̂ · radial, falls off as 1/r²

F = qE · the force on a test charge q

Relative field strength · 47%

Near · strong field ·

The electric field — a charge's map of force

When a single charge is present, every point around it gets an arrow saying how much force a unit positive charge placed there would feel. The collection of these arrows is the electric field E. Whether or not a test charge is actually there, the field is already spread around the charge. The test charge is assumed small enough not to disturb the source. So the electric field is defined as the force divided by the test charge, E = F/q.

A point charge's field — radial, 1/r²

The electric field of a point charge Q spreads equally in every direction, and its magnitude is inversely proportional to the square of the distance. As a formula, E = kQ/r², pointing along the radial unit vector r̂. It points outward for a positive charge and inward for a negative one. This inverse square arises because the field leaving the charge spreads evenly over a sphere whose area grows with the square of the distance. The constant k is 1/(4πε₀).

ObserveE = k Qr² r̂

The field of a point charge is proportional to the charge Q and inversely proportional to r², the square of the distance.

Force comes from the field — F = qE

A charge q placed in an electric field feels the force F = qE, the field arrow multiplied by the amount of charge. The key point is that the electric field E does not depend on the test charge q. Doubling q leaves E at that spot unchanged and only doubles the force. A negative charge feels a force opposite to the field. The electric field is a property of the source charge that created it.

ChooseF = ?

The force is the charge times the electric field.

Direction and adding up

Electric field arrows come out of positive charges and go into negative charges. With several charges, you add the arrows each one creates as vectors to get the field at a point; this is superposition. When charge is spread over a line, a surface, or a volume, this sum becomes an integral, and that integral is the topic of the next unit.

Fill in|E| ∝ ?

When the distance doubles, the field shrinks to a quarter.

On your ownE = ? r̂

The constant k is 1/(4πε₀).

Back to the first screen

As you moved the test charge closer with the slider, the force arrow grew sharply. Since the field goes as 1/r², halving the distance makes it four times stronger, and the relative strength on the indicator moved in the same ratio. Pressing the sign button flipped every arrow at once, which means the direction follows the sign of the source charge. And wherever you placed the test charge, the background field arrows stayed the same, because the electric field belongs not to the charge placed there but to the charge that created it.

The electric field E is a vector field that a charge creates in the space around it. The arrow at a point equals the force a unit positive charge placed there would feel (E = F/q). For a point charge Q, E = kQ/r² r̂ spreads in the radial direction and is inversely proportional to the square of the distance. The force on a charge q is F = qE.

What comes next

The electric field is the protagonist of electrostatics, and from here it branches two ways. When charge is spread over a line, a surface, or a volume, you integrate the dE from each small piece to get the whole field (EM-04). And counting how much of this field passes through a closed surface leads to flux and divergence (EM-05) and Gauss's law (EM-06).