Electric Field and Field Intensity: MDCAT Physics notes

Electric field

An electric field is the region around a charge in which another charge experiences an electric force. The electric field intensity (field strength) at a point is the force per unit positive test charge placed there:

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

• It is a vector. Its direction is the direction of the force on a positive charge.
• For a positive charge $\vec{F}$ and $\vec{E}$ are parallel; for a negative charge they are opposite.
• Units: N C$^{-1}$, which equals V m$^{-1}$ and J C$^{-1}$ m$^{-1}$. The J C$^{-1}$ (volt) alone is the unit of potential, not field.

Field belongs to the source

$E$ at a point is set by the source charges, not by the test charge. Placing a different test charge changes the force $F = qE$ but not $E$.

Example: in a field of 300 N C$^{-1}$, a charge of $-2$ C feels a force of magnitude $2\times300 = 600$ N (directed opposite to $E$), while $E$ stays 300 N C$^{-1}$.

Ratio of fields at two points using the same test charge equals the ratio of forces: 50 N and 150 N give $E_A:E_B = 1:3$.

Field of a point charge

$$E = \frac{1}{4\pi\varepsilon_0}\frac{q}{r^2}$$

It points away from a positive charge and towards a negative charge. $E \propto q$, so doubling every source charge doubles the resultant field at any point.

Two charges

At a point near two charges, the larger and nearer charge usually dominates the direction of the net field.

Uniform field and motion of a charge

Between two oppositely charged parallel plates the field is uniform:

$$E = \frac{V}{d}$$

A charged particle in it feels a constant force $F = qE$, so its acceleration $a = qE/m$ is constant (its velocity keeps changing). To find the force you need the charge, the potential difference and the plate separation; the particle's speed is not needed.

Field inside conductors and a medium

• Inside a charged hollow conducting sphere, $E = 0$. The charge resides on the outer surface.
• In a dielectric, the field is reduced: $E_{\text{med}} = E_{\text{vac}}/\varepsilon_r$.

Key formulas

• $E = F/q_0$
• $E = kq/r^2$
• $E = V/d$ (parallel plates)
• $a = qE/m$

Common MDCAT traps

• J C$^{-1}$ is the volt, a unit of potential, not field intensity.
• For unlike charges there is no zero-field point between them.
• Changing the test charge changes the force, not the field.
• In a uniform field acceleration is constant, not velocity.
• "Force per unit positive test charge" defines field strength, not potential.

Quick revision

• $E$ is a vector; unit N C$^{-1}$ = V m$^{-1}$.
• Field between parallel plates is uniform.
• Field inside a charged hollow sphere is zero.
• Midway between equal like charges, $E = 0$.
• A dielectric reduces $E$ by $\varepsilon_r$.