Class 12 Physics · NCERT notes for NEET

Electric Charges and Fields: NCERT Notes, Formulas and Tricks for NEET

Charge and its properties, Coulomb's law, the electric field and field lines, the dipole, and Gauss's law with its standard results. It is the base for all of electrostatics, and NEET regularly asks on dipoles, flux and Gauss's law results.

• 1. Electric charge and its properties
• 2. Coulomb's law and superposition
• 3. Electric field
• 4. Electric field lines
• 5. Electric dipole
• 6. Electric flux and Gauss's law
• 7. Applications of Gauss's law
• Quick revision

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Topic 1

Electric charge and its properties

Pull off a woollen sweater on a dry winter night and you hear it crackle: you have just moved billions of electrons, but you have not created a single charge.

• There are two kinds of charge, positive and negative. Like charges repel and unlike charges attract. Naming the charge on a glass rod rubbed with silk as positive is a convention set by Franklin.
• Charging by rubbing (friction) moves electrons from one body to another; it does not create charge. Glass rubbed with silk becomes positive (it loses electrons), and the silk becomes negative.
• Conductors, such as metals, the human body and the Earth, let charge move freely. Insulators, such as glass, plastic and rubber, do not. Earthing (grounding) means sharing a body's charge with the Earth.
• Charging by induction: bring a charged rod near a conductor, earth the conductor, break the earth connection, then remove the rod. The conductor ends up with charge opposite to the rod's, and the rod does not lose any charge.
• A charged body can attract a neutral body because it induces (polarises) opposite charge on the nearer side.
• Additivity: the total charge is the algebraic sum of the individual charges. Charge is a scalar.
• Conservation: the total charge of an isolated system stays constant. Charges can be created or destroyed only in equal and opposite pairs, as in pair production.
• Quantisation: $q=ne$ with $n$ an integer and $e=1.602\times 10^{-19}\text{C}$. At large scales charge can be treated as continuous.
• One coulomb is about $6.25\times 10^{18}$ electrons' worth of charge. Charge is also invariant: it does not depend on the speed of the body.
• A body's mass changes very slightly on charging. A negatively charged body gains electrons and so gains a tiny mass.

Formulas

• $q=\pm ne$
  Quantisation; $n$ is an integer, $e=1.6\times 10^{-19}$ C
• $n=\frac{q}{e}$
  Number of electrons transferred to give charge $q$
• $q^{\prime }=\frac{q_1+q_2}{2}$
  Charge on each of two identical conductors after they touch (signs included)

Tips, tricks and hacks

Common mistakes

NEET traps

Rubbing transfers electrons; it does not create charge. Only electrons move, not protons.

Quantisation is in units of $e$ for free particles. Quarks carry $\pm e/3$ and $\pm 2e/3$, but they are never found free.

Attraction can happen between a charged body and a neutral one, but repulsion only happens between two charged bodies. Repulsion is the sure test of charge.

Topic 2

Coulomb's law and superposition

The electric pull between an electron and a proton is about $10^{39}$ times their gravitational pull. Gravity rules the planets only because big bodies are almost perfectly neutral.

• Coulomb's law: the force between two point charges is $F=\frac{1}{4\pi {\epsilon }_0}\frac{q_1{q}_2}{r^2}$. It acts along the line joining them, repulsive for like charges and attractive for unlike charges.
• $\frac{1}{4\pi {\epsilon }_0}=9\times 10^9{\text{N m}}^2{\text{C}}^{-2}$, and ${\epsilon }_0=8.854\times 10^{-12}{\text{C}}^2{\text{N}}^{-1}{\text{m}}^{-2}$, the permittivity of free space.
• Vector form: ${\vec{F}}_{21}=\frac{1}{4\pi {\epsilon }_0}\frac{q_1{q}_2}{r_{21}^2}{\hat{r}}_{21}$, and ${\vec{F}}_{12}=-{\vec{F}}_{21}$. The forces obey Newton's third law.
• In a medium, $F_m=F_0/K$, where $K={\epsilon }_r=\epsilon /{\epsilon }_0$ is the dielectric constant. For water $K\approx 80$.
• One coulomb is the charge that repels an equal charge 1 m away in vacuum with a force of $9\times 10^9\text{N}$.
• Superposition: the net force on a charge is the vector sum of the forces from each other charge, each calculated as if the others were absent.
• The electric force between an electron and a proton is about $10^{39}$ times their gravitational attraction.
• Coulomb's law holds for point charges at rest. It is an inverse-square law, like gravitation, but the force can be attractive or repulsive.

Formulas

• $F=\frac{1}{4\pi {\epsilon }_0}\frac{q_1{q}_2}{r^2}$
  $\frac{1}{4\pi {\epsilon }_0}=9\times 10^9{\text{N m}}^2{\text{C}}^{-2}$; q in C, r in m
• ${\vec{F}}_{21}=\frac{1}{4\pi {\epsilon }_0}\frac{q_1{q}_2}{r_{21}^2}{\hat{r}}_{21}$
  Vector form; force on 2 due to 1, ${\hat{r}}_{21}$ points from 1 to 2
• $F_{\text{medium}}=\frac{F_{\text{vacuum}}}{K}$
  $K=\epsilon /{\epsilon }_0$, medium filling all space
• $\vec{F}={\vec{F}}_1+{\vec{F}}_2+\dots +{\vec{F}}_n$
  Superposition: vector sum of the forces from each charge
• $F_{\text{net}}=2F\cos (\phi /2)$
  Two equal forces $F$ at angle $\phi$
• $x=\frac{r}{1+\sqrt{q_2/q_1}}$
  Null point between two LIKE charges, measured from $q_1$

Tips, tricks and hacks

Common mistakes

NEET traps

Two identical metal spheres that touch share their total charge equally. For example, $+q$ and $-3q$ become $-q$ each. Recompute the force using the new charges.

$F$ drops by a factor $K$ only if the medium fills all the space. A slab of thickness $t$ placed between the charges gives $r_{\text{eff}}=r-t+t\sqrt{K}$.

A third charge placed on the line between two charges can be in equilibrium, but its equilibrium is not always stable.

Topic 3

Electric field

A charge does not need to touch anything to push it. It fills the space around it with a field, and any charge that wanders in feels the push.

• Electric field is the force per unit positive test charge, $\vec{E}={\lim }_{q_0\to 0}\vec{F}/q_0$. The test charge is made very small so that it does not disturb the source charges.
• SI unit of electric field: N/C, which is the same as V/m. It is a vector, and the force on a charge $q$ is $\vec{F}=q\vec{E}$.
• Field of a point charge: $E=\frac{1}{4\pi {\epsilon }_0}\frac{q}{r^2}$. It points radially outward for $+q$ and inward for $-q$. It has the same magnitude everywhere on a sphere around the charge.
• The field from several charges is the vector sum of the individual fields (superposition).
• The field is a physically real quantity, not just a calculation tool. In time-varying situations it carries energy and momentum, and disturbances in it travel at speed $c$.
• In a uniform field $E$, a charge accelerates with $a=qE/m$. An electron and a proton in the same field feel the same force but very different accelerations.
• An electron entering at right angles to a uniform field follows a parabolic path, like a projectile.
• NCERT example: an electron falling 1.5 cm in a field of $2\times 10^4\text{N/C}$ takes much less time than a proton over the same distance, because $t=\sqrt{2hm/(eE)}$. Gravity can be ignored.

Formulas

• $\vec{E}={\lim }_{q_0\to 0}\frac{\vec{F}}{q_0}$
  Definition with a small positive test charge
• $E=\frac{1}{4\pi {\epsilon }_0}\frac{q}{r^2}$
  Point charge; unit N/C = V/m
• $\vec{F}=q\vec{E}$
  Force on any charge $q$; opposite to $\vec{E}$ if $q<0$
• $a=\frac{qE}{m}$
  Acceleration in a uniform field
• $y=\frac{qE{x}^2}{2m{v}_x^2}$
  Sideways deflection after travelling $x$ across the field with speed $v_x$
• $t=\sqrt{\frac{2ym}{qE}}$
  Time to move $y$ from rest along the field

Tips, tricks and hacks

Common mistakes

NEET traps

The field at a point does not depend on the test charge placed there. The force on it does.

The field at the centre of a ring or a symmetric arrangement of equal charges is zero. The potential there is generally not zero.

In the same field, an electron's acceleration is about 1836 times a proton's, because its mass is that much smaller.

Topic 4

Electric field lines

Faraday was not strong in mathematics, so he drew 'lines of force' instead. Those sketches became the field idea that Maxwell later turned into equations.

• Field lines are curves drawn so that the tangent at any point gives the direction of $\vec{E}$ at that point.
• Field lines begin on positive charges and end on negative charges. For a single charge they start or end at infinity.
• Lines are drawn closer together where the field is strong and farther apart where it is weak. The number of lines through a unit perpendicular area is proportional to $E$.
• In a charge-free region, field lines are continuous curves with no breaks.
• Two field lines never cross. If they did, the field would have two directions at one point.
• Electrostatic field lines do not form closed loops, because the electrostatic field is conservative.
• A uniform field is shown by parallel, equally spaced straight lines. Lines meet a conductor's surface at right angles.
• Solid angle: $d\Omega =\Delta S/r^2$. The number of lines through a given solid angle stays the same with distance, which explains why $E\propto 1/r^2$.

Formulas

• $E\propto \frac{\Delta N}{\Delta S_{\perp }}$
  Line density (lines per unit perpendicular area) measures field strength
• $\Delta \Omega =\frac{\Delta S}{r^2}$
  Solid angle; fixed lines through a fixed solid angle spread over $r^2$, so $E\propto 1/r^2$

Tips, tricks and hacks

Common mistakes

NEET traps

Field lines are not the path of a charged particle (except in special cases, such as a charge starting from rest in a uniform field).

Electrostatic lines never form closed loops, but magnetic field lines always do.

Field lines never cross, and they never pass through the inside of a conductor in electrostatics.

Topic 5

Electric dipole

A water molecule is a tiny dipole. That is why a charged plastic comb bends a thin stream of tap water towards itself.

• An electric dipole is a pair of equal and opposite charges $\pm q$ a distance $2a$ apart. Its dipole moment is $\vec{p}=q\cdot 2a\hat{p}$, pointing from $-q$ to $+q$. SI unit: C m.
• Axial (end-on) field: $E=\frac{1}{4\pi {\epsilon }_0}\frac{2pr}{(r^2-a^2{)}^2}\approx \frac{2p}{4\pi {\epsilon }_0{r}^3}$ for $r\gg a$. It points along $\vec{p}$.
• Equatorial (broadside-on) field: $E=\frac{1}{4\pi {\epsilon }_0}\frac{p}{(r^2+a^2{)}^{3/2}}\approx \frac{p}{4\pi {\epsilon }_0{r}^3}$. It points opposite to $\vec{p}$.
• At the same large distance, the axial field is twice the equatorial field. A dipole's field falls as $1/r^3$, faster than a point charge's $1/r^2$.
• Ideal (point) dipole: $q\to \infty$ and $2a\to 0$ while $p$ stays finite.
• Polar molecules such as ${\text{H}}_2\text{O}$ and HCl have a permanent dipole moment. Non-polar molecules such as ${\text{CO}}_2$ and ${\text{CH}}_4$ have no permanent dipole moment, but an applied field can induce one.
• In a uniform field the net force on a dipole is zero, but it feels a torque $\vec{\tau }=\vec{p}\times \vec{E}$, with magnitude $pE\sin \theta$. The torque turns $\vec{p}$ to line up with $\vec{E}$.
• In a non-uniform field a dipole feels a net force as well. A dipole parallel to the field is pulled towards the stronger-field region, which is why a charged comb attracts bits of paper.

Formulas

• $p=q\times 2a$
  Dipole moment in C m, from $-q$ to $+q$
• $E_{\text{axial}}=\frac{1}{4\pi {\epsilon }_0}\frac{2pr}{(r^2-a^2{)}^2}\approx \frac{2p}{4\pi {\epsilon }_0{r}^3}$
  Along $\vec{p}$; approximation for $r\gg a$
• $E_{\text{eq}}=\frac{1}{4\pi {\epsilon }_0}\frac{p}{(r^2+a^2{)}^{3/2}}\approx \frac{p}{4\pi {\epsilon }_0{r}^3}$
  Opposite to $\vec{p}$
• $\vec{\tau }=\vec{p}\times \vec{E},\tau =pE\sin \theta$
  Torque in a uniform field; net force is zero
• $E=\frac{p}{4\pi {\epsilon }_0{r}^3}\sqrt{3{\cos }^2\theta +1}$
  Short dipole, general point at angle $\theta$ from the axis

Tips, tricks and hacks

Common mistakes

NEET traps

In a uniform field the net force on a dipole is zero but the torque is not, unless $\theta =0$ or $\pi$.

The equatorial field points opposite to $\vec{p}$. It is not zero.

The torque is maximum, $pE$, at $\theta =90^{\circ }$, and zero at $0^{\circ }$ (stable equilibrium) and $180^{\circ }$ (unstable equilibrium).

Topic 6

Electric flux and Gauss's law

Hold a hoop in the rain: face it upward and many drops pass through; tilt it edge-on and none do. Electric flux works the same way, with field lines in place of raindrops.

• Electric flux through a small area: $\Delta \varphi =\vec{E}\cdot \Delta \vec{S}=E\Delta S\cos \theta$. The area vector points along the normal, and outward for a closed surface.
• Flux is a scalar with unit N m²/C (equivalently V m). It is proportional to the number of field lines crossing the surface.
• Gauss's law: $\oint \vec{E}\cdot d\vec{S}=q_{\text{enc}}/{\epsilon }_0$. The total flux through any closed surface depends only on the net charge inside it.
• Charges outside the Gaussian surface add zero net flux. They do contribute to $\vec{E}$ at points on the surface.
• Gauss's law holds for any closed surface of any shape and size. It follows from the inverse-square nature of Coulomb's law.
• Choose the Gaussian surface to match the symmetry, so that $E$ is constant on it and perpendicular or parallel to it. The surface must not pass through a discrete point charge.
• A charge $q$ at the centre of a cube gives a flux of $q/{\epsilon }_0$ through the whole cube and $q/(6{\epsilon }_0)$ through each face.
• A closed surface with a dipole, or with no charge, inside has zero net flux through it.
• In a uniform field, the net flux through any closed surface is zero. For example, a cylinder whose axis is along $\vec{E}$ has inflow $-E\pi r^2$ and outflow $+E\pi r^2$.

Formulas

• $\varphi =\vec{E}\cdot \vec{S}=ES\cos \theta$
  Flat area in a uniform field; $\theta$ from the normal
• $\oint \vec{E}\cdot d\vec{S}=\frac{q_{\text{enc}}}{{\epsilon }_0}$
  Gauss's law; algebraic sum of enclosed charge
• ${\varphi }_{\text{face}}=\frac{q}{6{\epsilon }_0}$
  Each face of a cube with $q$ at its centre
• ${\varphi }_{\text{cube}}=\frac{q}{2{\epsilon }_0},\frac{q}{8{\epsilon }_0}$
  Charge at a face centre, at a corner
• ${\varphi }_{\text{medium}}=\frac{q}{K{\epsilon }_0}$
  Charge in a medium of dielectric constant $K$
• $[\varphi ]=[{\text{M L}}^3{\text{T}}^{-3}{\text{A}}^{-1}]$
  Unit N m${}^2$ C${}^{-1}$ = V m

Tips, tricks and hacks

Common mistakes

NEET traps

The total flux does not depend on the size or shape of the surface, or on where the charge sits inside it. The field at points on the surface does change.

$\theta$ is the angle with the NORMAL to the surface, not with the surface itself.

A charge at a cube's corner sends $q/(8{\epsilon }_0)$ through the cube. A charge at the centre of a face sends $q/(2{\epsilon }_0)$ through it.

Topic 7

Applications of Gauss's law

Gauss's law turns pages of integration into one line, but only when the charge has symmetry: a long wire, a big flat sheet or a sphere.

• Infinitely long straight wire with linear charge density $\lambda$: use a cylindrical Gaussian surface. $E=\frac{\lambda }{2\pi {\epsilon }_{0}r}$, directed radially, so $E\propto 1/r$.
• Infinite plane sheet with surface charge density $\sigma$: use a pillbox (cylinder) through the sheet. $E=\frac{\sigma }{2{\epsilon }_0}$, the same at every distance and normal to the sheet.
• Two parallel sheets with $+\sigma$ and $-\sigma$: the field between them is $\sigma /{\epsilon }_0$ and the field outside is zero. This is the field in a parallel plate capacitor.
• Thin spherical shell of charge $q$ and radius $R$: outside ($r>R$) the field is $\frac{q}{4\pi {\epsilon }_0{r}^2}$, as if all the charge were at the centre. Inside ($r<R$) the field is $E=0$.
• Just outside the shell, $E=\sigma /{\epsilon }_0$, where $\sigma =q/(4\pi R^2)$. The field jumps from 0 to this value as you cross the shell.
• Uniformly charged solid sphere (a non-conductor): inside, $E=\frac{qr}{4\pi {\epsilon }_0{R}^3}$, which increases linearly with $r$. Outside it follows the point-charge law.
• Outside a charged conductor, $E=\sigma /{\epsilon }_0$, normal to the surface. Inside the conductor's material, $E=0$.
• These results work only because of symmetry: infinite length, infinite area, or spherical symmetry. For a finite wire or sheet they are only approximations near the middle.

Formulas

• $E=\frac{\lambda }{2\pi {\epsilon }_{0}r}$
  Infinite line charge. lambda is in C/m.
• $E=\frac{\sigma }{2{\epsilon }_0}$
  Infinite thin sheet of charge, independent of distance. sigma is in C/m^2.
• $E=\frac{1}{4\pi {\epsilon }_0}\frac{q}{r^2}(r>R),E=0(r<R)$
  Thin spherical shell of charge.
• $E=\frac{\sigma }{{\epsilon }_0}$
  Just outside a charged conductor, or between oppositely charged parallel plates.
• $E=\frac{1}{4\pi {\epsilon }_0}\frac{qr}{R^3}(r<R)$
  Inside a uniformly charged solid (non-conducting) sphere.

Tips, tricks and hacks

Common mistakes

NEET traps

A thin sheet of charge gives $\sigma /(2{\epsilon }_0)$, but just outside a conductor the field is $\sigma /{\epsilon }_0$. Do not mix the two.

Inside a charged shell $E=0$, but the potential there is not zero. It is constant and equal to $\frac{q}{4\pi {\epsilon }_{0}R}$.

For a line charge $E\propto 1/r$. For a point charge $E\propto 1/r^2$. For a sheet, $E$ does not depend on $r$.

Quick revision: Electric Charges and Fields the night before NEET

1. $q=ne$, $e=1.6\times 10^{-19}$ C; charge is additive, conserved, quantised and invariant.
2. Charging by rubbing or induction only moves electrons; in induction the rod loses no charge.
3. Identical conductors that touch share the total charge equally: $(q_1+q_2)/2$ each.
4. $F=\frac{1}{4\pi {\epsilon }_0}\frac{q_1{q}_2}{r^2}$, $\frac{1}{4\pi {\epsilon }_0}=9\times 10^9$ N m${}^2$ C${}^{-2}$; in a medium $F/K$.
5. Two equal forces at angle $\phi$: $2F\cos (\phi /2)$; equilateral triangle of charges gives $\sqrt{3}F$.
6. $E=F/q_0$ in N/C = V/m; force on an electron is opposite to $\vec{E}$.
7. Same field: $a_e/a_p\approx 1836$.
8. Field lines: start on $+$, end on $-$, never cross, never form closed loops, meet conductors at $90^{\circ }$.
9. $\vec{p}$ from $-q$ to $+q$; $E_{\text{axial}}=2kp/r^3$ along $\vec{p}$, $E_{\text{eq}}=kp/r^3$ opposite to $\vec{p}$.
10. $\tau =pE\sin \theta$; net force zero in a uniform field; $0^{\circ }$ stable, $180^{\circ }$ unstable.
11. $\varphi =ES\cos \theta$ with $\theta$ from the normal; unit N m${}^2$/C = V m.
12. Gauss: $\oint \vec{E}\cdot d\vec{S}=q_{\text{enc}}/{\epsilon }_0$; outside charges add zero net flux.
13. Cube: centre $q/(6{\epsilon }_0)$ per face; face centre $q/(2{\epsilon }_0)$; corner $q/(8{\epsilon }_0)$ total.
14. Line $\lambda /(2\pi {\epsilon }_{0}r)$; sheet $\sigma /(2{\epsilon }_0)$; conductor or between plates $\sigma /{\epsilon }_0$.
15. Shell: $E=0$ inside, $kq/r^2$ outside; solid sphere inside $E=kqr/R^3$.

Practise Electric Charges and Fields questions

Topic-wise practice from Level 1 to Level 3, NCERT-based questions and NEET PYQs, with a full solution after every answer.