Physics · Ch 1 — Electrostatics
Electric Field
Electric Field
To describe how a source charge q influences the space around it, physicists define a quantity called the electric field intensity, or simply the electric field E, at every point in that space. The electric field at a point P, a distance r from the source charge q, is defined as the force that would be experienced by a small positive test charge q0 placed at P, divided by the magnitude of that test charge: E = F/q0 = k q/r^2 r-hat = q/(4 pi epsilon0 r^2) r-hat, where r-hat is the unit vector pointing from q toward P. The electric field is a vector quantity, with SI unit newton per coulomb (N C^-1). Several important points follow from this definition. If the source charge q is positive, E points radially away from q; if q is negative, E points radially toward q. Once the field E at a point is known, the force on any test charge q0 placed there follows immediately as F = q0 E -- this restates Coulomb's law entirely in terms of the field, without needing to refer back to the source charge directly. The field at a point is completely determined by the source charge alone and does not depend on the (idealised, vanishingly small) test charge used to probe it; the test charge is taken small enough that bringing it in does not disturb the source charge's own field. Because the electric field is a vector, it has a unique magnitude and direction at every point in space; as distance from a point-charge source increases, the field magnitude decreases according to the inverse-square law. A field is called uniform if it has the same magnitude and direction everywhere in a region (produced, for instance, by an infinite charged plane, as later sections show), and non-uniform if its magnitude or direction (or both) vary from point to point -- the field of an isolated point charge is a standard example of a non-uniform field, since its direction is always radial and its magnitude always falls off with distance. The point-charge formula for E is strictly valid only for po …
What this figure shows. Two side-by-side diagrams show a source charge at the centre with arrows radiating outward in all directions from a positive source charge, and arrows converging inward toward a negative source charge, with a field-point P marked at the same distance r in each case. The picture encodes the rule that a positive source charge's field points radially outward (away from the charge) at every surrounding point, while a negative source charge's field points radially inward (toward the charge), which follows directly from the direction a small positive test c …
What this figure shows. A test charge q0 is shown placed at point P within the field E created by a source charge q; an arrow labelled F = q0 E shows the force the test charge experiences, directed the same way as E when q0 is positive (away from a positive source, toward a negative source) and opposite to E when q0 is negative. This is the diagram that connects the abstract field vector E back to something measurable and concrete -- the actual mechanical force felt by any real …
Worked out. For a positive point charge +1 uC placed at the origin, with point P at 4 m along the y-axis... and point Q at 2 m further along, the field magnitude at P works out to E_P = k q/r_P^2 = 2.25x10^3 NC^-1 directed away from the positive source (along +x at P), and similarly E_Q is computed at Q's distance using the same inverse-square formula. For a negative point charge -2 uC placed at the origin, the field magnitude at P (distance 6 m) works out to E_P = 4.5x10^3 NC^-1, but since the source is negative the field points toward the charge, i.e. in the -x direction at P, while at the more distant point Q the field is E_Q = 0.5x10^3 NC^-1 pointing in the +x direction (toward the negative source from Q's side). Both cases show the same inverse-square falloff with distance …
What this figure shows. Field vectors EP, EQ and ER are drawn at three different points P, Q, R around a positive source charge in panel (a), and around a negative source charge in panel (b), with the arrow lengths deliberately drawn unequal -- longest at the point closest to the source charge and shortest at the point farthest away -- to make visually clear that the field's magnitude falls off as the field point moves farther from the source, exactly as the inverse-square formula E = kq/r^2 predicts. …
What this figure shows. Several small regions of space are shown side by side: one filled with parallel, evenly-spaced field-line arrows all pointing the same direction and having the same length everywhere (a uniform field), and three others filled with field lines that either fan out, curve, or change spacing from point to point (non-uniform fields). The figure makes the distinction concrete -- a uniform field has both the same direction and the same magnitude at every point in the region, while a non-uniform field, such as the field of an isolated point charge, changes in direction, …