Electric Field — Revision Notes

⚡ 30-Second Revision

Definition: — $\vec{E} = \frac{\vec{F}}{q_0}$ (Force per unit positive test charge).
Unit: — N/C or V/m.
Point Charge: — $E = \frac{1}{4\pi\epsilon_0} \frac{|Q|}{r^2}$. Direction: away from positive, towards negative.
Superposition: — $\vec{E}_{net} = \sum \vec{E}_i$ (vector sum).
Field Lines: — Originate from +, terminate on -. Never intersect. Tangent gives direction. Density = strength. No closed loops.
Infinite Line: — $E = \frac{\lambda}{2\pi\epsilon_0 r}$.
Infinite Sheet: — $E = \frac{\sigma}{2\epsilon_0}$.
Charged Ring (axis): — $E_x = \frac{1}{4\pi\epsilon_0} \frac{Qx}{(R^2 + x^2)^{3/2}}$.
Dipole Torque (uniform E): — $\vec{\tau} = \vec{p} \times \vec{E}$, magnitude $\tau = pE \sin\theta$.
Dipole Potential Energy (uniform E): — $U = -\vec{p} \cdot \vec{E}$, magnitude $U = -pE \cos\theta$.
Dipole in non-uniform E: — Experiences both net force and torque.

2-Minute Revision

The electric field ($\vec{E}$) is a vector quantity representing the electric force per unit positive test charge. It's the influence a source charge creates in space, measured in N/C or V/m. For a point charge $Q$, the field magnitude is $E = k|Q|/r^2$, directed radially away from positive $Q$ and towards negative $Q$.

When multiple charges are present, the net field is found by vectorially adding individual fields (superposition principle). Electric field lines are visual aids: they start on positive charges, end on negative charges, never intersect, and their density indicates field strength.

Electrostatic field lines do not form closed loops. For continuous charge distributions, remember key results: $E = \frac{\lambda}{2\pi\epsilon_0 r}$ for an infinite line and $E = \frac{\sigma}{2\epsilon_0}$ for an infinite plane sheet.

An electric dipole (moment $\vec{p}$) in a uniform electric field $\vec{E}$ experiences a torque $\vec{\tau} = \vec{p} \times \vec{E}$ and has potential energy $U = -\vec{p} \cdot \vec{E}$. In a non-uniform field, a dipole experiences both a net force and a torque.

5-Minute Revision

The electric field, $\vec{E}$, is a fundamental concept in electrostatics, quantifying the electric influence at any point in space. It's defined as the force experienced by a vanishingly small positive test charge ($q_0$) per unit charge: $\vec{E} = \vec{F}/q_0$.

The SI unit is N/C or V/m. For a single point charge $Q$, the magnitude of the electric field at a distance $r$ is $E = \frac{1}{4\pi\epsilon_0} \frac{|Q|}{r^2}$. The direction is radially outward for a positive source charge and inward for a negative one.

Remember $k = 1/(4\pi\epsilon_0) = 9 \times 10^9\text{ N m}^2/\text{C}^2$.

When dealing with multiple point charges, the net electric field at any point is the vector sum of the fields produced by each individual charge (Principle of Superposition). This requires careful vector addition, often by resolving fields into x and y components. For example, if two charges $Q_1$ and $Q_2$ are present, $\vec{E}_{net} = \vec{E}_1 + \vec{E}_2$.

Electric field lines are a graphical representation. Key properties to recall: they originate from positive charges and terminate on negative charges (or infinity); they never intersect; the tangent to a field line gives the direction of $\vec{E}$; their density indicates field strength; and they do not form closed loops. These properties are frequently tested conceptually.

For continuous charge distributions, specific formulas are important: for an infinite line of charge with linear density $\lambda$, $E = \frac{\lambda}{2\pi\epsilon_0 r}$; for an infinite plane sheet with surface density $\sigma$, $E = \frac{\sigma}{2\epsilon_0}$.

Note that for the infinite sheet, the field is uniform and independent of distance. For a uniformly charged ring of radius $R$ and charge $Q$, at a distance $x$ along its axis, $E_x = \frac{1}{4\pi\epsilon_0} \frac{Qx}{(R^2 + x^2)^{3/2}}$.

An electric dipole, characterized by its dipole moment $\vec{p}$, behaves distinctly in an electric field. In a uniform electric field $\vec{E}$, it experiences a torque $\vec{\tau} = \vec{p} \times \vec{E}$ (magnitude $\tau = pE \sin\theta$) that tends to align it with the field.

Its potential energy is $U = -\vec{p} \cdot \vec{E}$ (magnitude $U = -pE \cos\theta$), with minimum energy when aligned ($\theta=0^\circ$) and maximum when anti-aligned ($\theta=180^\circ$). Crucially, in a non-uniform electric field, the dipole experiences both a net force and a torque, as the forces on its two charges are no longer equal and opposite.

Prelims Revision Notes

Electric Field ($\vec{E}$)

Definition: — Electric force per unit positive test charge. $\vec{E} = \frac{\vec{F}}{q_0}$.
Units: — N/C or V/m.
Nature: — Vector quantity. Direction is same as force on positive test charge.

Electric Field due to Point Charge

Magnitude: — $E = \frac{1}{4\pi\epsilon_0} \frac{|Q|}{r^2} = k \frac{|Q|}{r^2}$.
Direction: — Radially outward from positive charge, radially inward towards negative charge.
Constant: — $k = 9 \times 10^9\text{ N m}^2/\text{C}^2$.

Principle of Superposition

For multiple charges, the net electric field at a point is the vector sum of the electric fields due to individual charges: $\vec{E}_{net} = \sum_{i} \vec{E}_i$.
Requires careful vector addition (component method).

Electric Field Lines (Lines of Force)

Origin/Termination: — Start from positive charges, end on negative charges (or infinity).
Non-intersection: — No two field lines ever intersect (unique field direction).
Direction: — Tangent to the line at any point gives the direction of $\vec{E}$.
Strength: — Density of lines (closeness) indicates field strength (denser = stronger).
Loops: — Do NOT form closed loops (electrostatic field is conservative).
Conductors: — Perpendicular to the surface of a conductor.

Electric Field due to Continuous Charge Distributions

General Formula: — $\vec{E} = \int d\vec{E} = \int \frac{1}{4\pi\epsilon_0} \frac{dQ}{r^2} \hat{r}$.
Infinite Line Charge (linear density $\lambda$): — $E = \frac{\lambda}{2\pi\epsilon_0 r}$ (radial).
Infinite Plane Sheet (surface density $\sigma$): — $E = \frac{\sigma}{2\epsilon_0}$ (uniform, perpendicular to sheet).
Uniformly Charged Ring (on axis, distance $x$ from center, radius $R$, charge $Q$): — $E_x = \frac{1}{4\pi\epsilon_0} \frac{Qx}{(R^2 + x^2)^{3/2}}$. At center ($x=0$), $E=0$. For $x \gg R$, approximates point charge.

Electric Dipole in Electric Field

Dipole Moment: — $\vec{p} = q(2\vec{a})$ (from $-q$ to $+q$).
Torque (uniform $\vec{E}$): — $\vec{\tau} = \vec{p} \times \vec{E}$. Magnitude $\tau = pE \sin\theta$.

* Max torque at $\theta = 90^\circ$. * Zero torque at $\theta = 0^\circ, 180^\circ$.

Potential Energy (uniform $\vec{E}$): — $U = -\vec{p} \cdot \vec{E} = -pE \cos\theta$.

* Minimum $U = -pE$ (stable equilibrium) at $\theta = 0^\circ$. * Maximum $U = +pE$ (unstable equilibrium) at $\theta = 180^\circ$.

Non-uniform $\vec{E}$: — Dipole experiences both a net force and a torque.

Vyyuha Quick Recall

To remember electric field line properties: NICE Never intersect. Infinity (or negative charge) is where they end. Close lines mean strong field. Emerge from positive charges.