Electric Charges and Fields — NEET notes

Electric Charge

The fundamental property of matter responsible for all electromagnetic phenomena. Charges come in two types (positive and negative), are conserved in every process, and exist only in integer multiples of the electron charge.

q = ne (n = 0, 1, 2, )

What is Electric Charge?

Electric charge is an intrinsic property of matter, just like mass. But unlike mass (which is always positive), charge comes in two flavours: positive and negative.

Electric charge is a fundamental property of matter that causes it to experience a force in an electric or magnetic field. Just like mass causes gravitational attraction, charge causes electromagnetic interaction.

• Two types of charge: positive (protons) and negative (electrons)
• Like charges repel, unlike charges attract
• SI unit of charge: coulomb (C)
• Charge of one electron: e = 1.6 10^ -19 C
• Charge of one proton: +1.6 10^ -19 C (equal magnitude, opposite sign)

Conservation of Charge

The total electric charge in an isolated system always remains constant. Charges can be transferred between bodies, but the algebraic sum never changes.

The law of conservation of charge states that the net electric charge of an isolated system remains constant. You can transfer charge from one body to another, but you cannot create or destroy it. This law holds in every known physical process — no exception has ever been found.

• Total charge before any process = Total charge after the process
• Rubbing doesn't create charge — it transfers electrons between surfaces
• In nuclear reactions: total charge of reactants = total charge of products
• Conservation of charge is as fundamental as conservation of energy

Quick recall

A neutral body is split into two pieces. One piece has charge +3 C. What is the charge on the other piece?

The other piece must have -3 C. The original body was neutral (total charge = 0), and charge is conserved, so +3 + (-3) = 0 .

Quantization of Charge

Electric charge is not continuous — it exists only in integer multiples of the elementary charge e. You can have 1e, 2e, 3e... but never 1.5e.

Quantization of charge means that charge always occurs in integer multiples of the elementary charge e = 1.6 10^ -19 C. You can have q = ne where n is any integer, but you can never have a fraction of e .

q = ne where n = 0, 1, 2, 3,

• Millikan's oil-drop experiment (1909) first proved charge quantization
• Quarks have charges e 3 and 2e 3 , but they are always confined inside protons/neutrons — free quarks don't exist
• At macroscopic scale, quantization is undetectable ( 1 C = 6.25 10^ 12 electrons)
• Quantization matters only at atomic/molecular scales

Methods of Charging

There are three main ways to charge a body: rubbing (friction), touching (conduction), and bringing a charged body near without touching (induction).

An uncharged (neutral) body has equal numbers of protons and electrons. To charge it, we must create an imbalance. There are three methods: friction (rubbing two different materials), conduction (touching a charged body to a neutral one), and induction (using a nearby charge to redistribute electrons without contact).

Charging by Induction (step-by-step)

1. Bring a charged rod (say positive) near a neutral conductor — electrons in the conductor are attracted toward the rod, leaving the far end positive
2. While the rod is still near, ground the conductor — electrons flow in from the ground to neutralize the far-end positive charge
3. Remove the ground connection first, then remove the rod — the conductor now has excess electrons and is negatively charged
4. Key: the induced charge is always OPPOSITE to the charge on the inducing body

• Friction: electrons transfer from one material to another (triboelectric series determines direction)
• Conduction: charge flows until both bodies reach the same potential
• Induction: no charge is transferred from the external body — electrons within the conductor rearrange
• Only electrons move in solid conductors — never protons

Quick recall

A positively charged rod touches a neutral metal sphere. What charge does the sphere acquire?

The sphere becomes positively charged. In conduction, charge flows from the charged body to the neutral body until equilibrium — both end up with the same sign (positive).

The triboelectric series (glass > hair > nylon > silk > paper > rubber > PVC) tells you which material loses electrons when rubbed against another. Higher in the series = more likely to become positive.

Coulomb's Law

The force between two point charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them. It's the electric equivalent of Newton's gravitational law — but about 10³⁶ times stronger.

F = 1 4 _0 q_1 q_2 r^2

Coulomb's Law

The fundamental law that quantifies the force between two stationary point charges. Discovered by Charles-Augustin de Coulomb in 1785 using a torsion balance.

Coulomb's Law states: the electrostatic force between two stationary point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. The force acts along the line joining the two charges.

F = 1 4 _0 |q_1||q_2| r^2

Here _0 = 8.854 10^ -12 C ^2 N ^ -1 m ^ -2 is the permittivity of free space. The constant k = 1 4 _0 = 9 10^9 Nm ^2 C ^ -2 is called Coulomb's constant.

• Force is attractive for unlike charges ( q_1 q_2 < 0 ), repulsive for like charges ( q_1 q_2 > 0 )
• Obeys Newton's Third Law: F _ 12 = - F _ 21
• Valid only for point charges (or spherical charge distributions at distances >> radius)
• In a medium with dielectric constant : F = 1 4 _0 q_1 q_2 r^2 — force decreases by factor

Vector Form

Coulomb's law in vector form explicitly gives both the magnitude and direction of the force, using the unit vector along the line joining the charges.

The vector form of Coulomb's law for the force on charge q_1 due to charge q_2 is:

F _ 12 = 1 4 _0 q_1 q_2 r_ 12 ^2 r _ 12

Here r _ 12 is the unit vector pointing from q_2 to q_1 . If both charges have the same sign, q_1 q_2 > 0 , and F _ 12 points along r _ 12 (repulsion). If opposite signs, q_1 q_2 < 0 , and the force is anti-parallel to r _ 12 (attraction).

• r _ 12 = unit vector from q_2 to q_1 (from source to target)
• Sign of q_1 q_2 automatically handles attraction/repulsion — no need to manually assign direction
• Magnitude: use |q_1||q_2| ; Direction: let the algebra handle the sign

Coulomb's Law vs Gravity

Both are inverse-square laws, but electrostatic force is enormously stronger than gravitational force at the atomic scale.

Superposition Principle

The net force on a charge due to multiple other charges equals the vector sum of individual Coulomb forces from each charge, calculated independently.

The principle of superposition states: the force on any charge due to a number of other charges is the vector sum of all the forces exerted on it by the individual charges. Each pairwise force is calculated independently using Coulomb's law, unaffected by the presence of other charges.

F _1 = F _ 12 + F _ 13 + F _ 14 + = _ i 1 F _ 1i

How to solve superposition problems (3+ charges)

1. Draw a clear diagram with all charges and distances labeled
2. Pick the charge on which you need the net force — call it the 'target' charge
3. Calculate the force from EACH other charge on the target using Coulomb's law (magnitude only first)
4. Find the direction of each force (attraction toward / repulsion away from the source charge)
5. Resolve all forces into x and y components: F_x = F , F_y = F
6. Add all x-components and all y-components separately: F_ net,x = F_x , F_ net,y = F_y
7. Find net magnitude: F_ net = F_x^2 + F_y^2 and direction: = ^ -1 (F_y/F_x)

Quick recall

Three equal charges q are placed at the corners of an equilateral triangle of side a . What is the net force on any one charge?

Each charge experiences two equal repulsive forces at 60° to each other. Magnitude of each: F = kq^2/a^2 . Net force: F_ net = F^2 + F^2 + 2F^2 60° = F 3 = 3 kq^2 a^2 , directed away from the center of the triangle.

Superposition is NOT obvious — it's a deep property of electromagnetism. It works because Maxwell's equations are linear. In quantum electrodynamics at extreme field strengths, superposition can break down (but not at NEET level!).

Electric Field

A charge creates an invisible 'influence zone' around it called the electric field. Any other charge placed in this zone feels a force. The field E at a point equals the force per unit positive test charge placed there.

E = F q_0 = 1 4 _0 q r^2 r

Electric Field Definition

The electric field at a point is defined as the force experienced by a small positive test charge placed at that point, divided by the magnitude of the test charge.

The electric field E at a point in space is defined as the electrostatic force F experienced by a small positive test charge q_0 placed at that point, divided by q_0 .

E = _ q_0 0 F q_0

The test charge q_0 must be infinitesimally small so it doesn't disturb the original charge distribution. The field exists at every point in space, whether or not a test charge is actually present — it's a property of the source charge.

• SI unit of E : N/C or equivalently V/m (volts per meter)
• E is a vector — it has both magnitude and direction
• Direction of E = direction of force on a positive test charge
• A negative charge experiences force opposite to E : F = q E (if q < 0 , F is anti-parallel to E )

Field Due to a Point Charge

A single point charge creates a radially symmetric electric field that decreases with the square of the distance.

The electric field at distance r from a point charge q is obtained by dividing Coulomb's force by the test charge q_0 :

E = 1 4 _0 q r^2 r

For a positive source charge, E points radially outward (away from the charge). For a negative source charge, E points radially inward (toward the charge). The magnitude falls off as 1/r^2 — double the distance, field drops to one-quarter.

• At the location of the charge itself ( r = 0 ), the field is undefined (infinite)
• The field is spherically symmetric — same magnitude at all points equidistant from q
• Field of a positive charge: radially outward; negative charge: radially inward
• The formula is identical to Coulomb's force with q_0 = 1 — that's the whole point of defining E

Superposition of Electric Fields

Just like forces, the net electric field at a point due to multiple charges equals the vector sum of the individual fields produced by each charge independently.

The superposition principle applies to electric fields just as it does to forces. The net electric field at any point is the vector sum of the fields produced by each individual charge, calculated as if the other charges weren't there.

E _ net = E _1 + E _2 + E _3 + = _i E _i

Solving for net field at a point P due to multiple charges

1. Find the distance from each charge to point P
2. Calculate the magnitude of E from each charge: E_i = kq_i/r_i^2
3. Determine the direction of each E _i (away from + charges, toward − charges)
4. Resolve each E _i into components along chosen axes
5. Sum components: E_x = E_ ix , E_y = E_ iy
6. Net field: E_ net = E_x^2 + E_y^2 , direction: = ^ -1 (E_y/E_x)

Quick recall

Two charges +q and -q are placed at distance 2a apart. What is the field at the midpoint?

Both fields point in the same direction (from +q toward -q ). Magnitude: E_ net = 2 kq a^2 = 2kq a^2 . The fields add up, not cancel, because both point the same way.

Electric Field Lines

Imaginary curves that visually represent the electric field. The tangent at any point gives the field direction, and the density of lines indicates field strength. A brilliant visualization tool invented by Michael Faraday.

Tangent at any point = Direction of E

What are Field Lines?

Electric field lines are imaginary curves drawn so that the tangent at each point gives the direction of the net electric field at that point.

Electric field lines (or lines of force) are curves drawn in a region of electric field such that the tangent to the curve at any point gives the direction of E at that point. They were introduced by Michael Faraday, who had no formal math training — he thought visually, and this tool made electromagnetic theory accessible.

The density (closeness) of field lines represents the strength of the field. Where lines are packed tightly, E is strong. Where they're spread out, E is weak. This gives an instant visual map of both direction and magnitude.

Properties of Field Lines

Field lines obey strict rules: they start on positive charges, end on negative charges, never cross, and are continuous curves (no breaks in empty space).

• Field lines start from positive charges and end on negative charges (or go to infinity)
• They never cross — if they did, there would be two field directions at the crossing point, which is impossible
• They never form closed loops in electrostatics (this distinguishes them from magnetic field lines)
• The number of lines leaving a charge is proportional to |q| — a +2q charge has twice as many lines as +q
• They are perpendicular to the surface of a conductor at every point
• Closer lines = stronger field; farther apart = weaker field

Field Line Patterns

Specific charge arrangements produce characteristic field line patterns that NEET expects you to recognize instantly.

Memorize these field line patterns — they appear in nearly every exam as 'identify the charge configuration' questions.

• Single +q : lines radiate outward uniformly in all directions
• Single -q : lines converge inward uniformly from all directions
• Dipole ( +q , -q ): lines curve from + to - ; a neutral point exists only at infinity along the perpendicular bisector
• Two equal +q : lines repel in the middle; there's a null point (E = 0) at the midpoint where fields cancel
• Unequal charges ( +4q , -q ): 4 times more lines leave +4q than enter -q ; some lines escape to infinity

Quick recall

Two charges +q and -3q are placed some distance apart. At a point far away, what does the field line pattern look like?

At very large distances, the system looks like a single charge of +q + (-3q) = -2q . Field lines converge inward as if there's a -2q charge. Close up, the pattern is complex — but far away, only the net charge matters.

Electric Dipole

An electric dipole is a pair of equal and opposite charges separated by a small distance. It's the simplest system beyond a single charge, and it models real molecules like H₂O. The dipole moment p = qd points from −q to +q.

p = q d (from -q to +q)

Dipole Moment

The electric dipole moment is a vector quantity that captures both the magnitude of the charges and the separation between them.

An electric dipole consists of two equal and opposite charges +q and -q separated by a distance 2a . The dipole moment p is defined as:

p = q 2a p (SI unit: C·m)

The direction of p is from the negative charge to the positive charge (from -q to +q ). This is a convention — don't confuse it with the field direction!

• Direction of p : from -q to +q (negative to positive)
• SI unit: coulomb-meter (C·m); often expressed in debye ( 1 D = 3.33 10^ -30 C·m)
• A dipole has zero net charge but a non-zero dipole moment
• Polar molecules (HCl, H₂O, NH₃) have permanent dipole moments

Field on the Axial Line

The electric field at a point on the dipole axis (the line joining the two charges, extended) is along the axis and proportional to 1/r³ at large distances.

For a point P on the axial line at distance r from the center of the dipole (with r 2a ), the electric field is:

E_ axial = 1 4 _0 2p r^3 (r a)

Direction: along p (from -q to +q , i.e., same direction as the dipole moment). The exact expression (without approximation) is:

E_ axial = 1 4 _0 2pr (r^2 - a^2)^2

• Axial field falls as 1/r^3 (faster than a point charge's 1/r^2 )
• Direction: parallel to p (same direction as dipole moment)
• This is the stronger field — axial field is twice the equatorial field at the same distance

Field on the Equatorial Line

The electric field at a point on the perpendicular bisector (equatorial plane) of the dipole is anti-parallel to the dipole moment and half the axial field.

For a point P on the equatorial line (perpendicular bisector) at distance r from the center of the dipole (with r 2a ):

E_ eq = 1 4 _0 p r^3 (r a)

Direction: opposite to p (anti-parallel — pointing from +q toward -q ). The exact expression is:

E_ eq = 1 4 _0 p (r^2 + a^2)^ 3/2

Torque & Energy of a Dipole in Uniform Field

A uniform electric field exerts a torque on a dipole, trying to align it with the field. This leads to stable and unstable equilibrium positions.

When a dipole is placed in a uniform electric field E , the forces on +q and -q are equal and opposite ( qE each), so the net force is zero. But these forces act at different points, creating a torque (turning effect).

= p E = pE

The torque tries to align p with E . The potential energy of the dipole in the field is:

U = - p E = -pE

• = 0° : p E , = 0 , U = -pE (minimum) → stable equilibrium
• = 180° : p anti-parallel to E , = 0 , U = +pE (maximum) → unstable equilibrium
• = 90° : = pE (maximum torque), U = 0
• Net force on dipole in uniform field = 0 (no translation, only rotation)
• In a non-uniform field: net force ≠ 0 (dipole translates toward stronger field region)

Quick recall

A dipole with moment p = 4 10^ -9 C·m is at 30° to a field E = 5 10^4 N/C. Find the torque.

= pE = 4 10^ -9 5 10^4 30° = 4 10^ -9 5 10^4 0.5 = 10^ -4 N·m.

Continuous Charge Distributions

Real objects aren't point charges — charge is spread over lines, surfaces, or volumes. We describe these using charge densities (λ, σ, ρ) and integrate to find the total field.

= dq dl , = dq dA , = dq dV

Three Types of Charge Density

When charge is distributed continuously over a wire, a sheet, or a solid body, we use charge density to describe how charge is spread out.

At the atomic scale, charge is discrete (electrons and protons). But for macroscopic objects with billions of charges, it's far more practical to treat charge as continuously spread. We define three types of charge density depending on the geometry.

• Linear charge density = dq/dl — charge per unit length (C/m). Use for wires, rods, rings
• Surface charge density = dq/dA — charge per unit area (C/m²). Use for plates, spherical shells, discs
• Volume charge density = dq/dV — charge per unit volume (C/m³). Use for solid spheres, cylinders, cubes

Finding Field by Integration

For continuous distributions, we divide the object into infinitesimal charge elements dq, find the field due to each, and integrate over the entire distribution.

To find the electric field of a continuous charge distribution, we use the superposition principle in integral form. Divide the distribution into infinitesimal elements dq , each treated as a point charge.

d E = 1 4 _0 dq r^2 r

E = d E = 1 4 _0 dq r^2 r

Strategy for continuous charge problems

1. Identify the type of distribution: dq = dl (line), dA (surface), or dV (volume)
2. Choose a convenient coordinate system (Cartesian, cylindrical, or spherical)
3. Write d E from a typical element dq at the field point
4. Exploit symmetry to eliminate components that cancel (e.g., perpendicular components of a ring cancel by symmetry)
5. Integrate over the entire distribution to get E

Field Due to a Charged Ring

The electric field on the axis of a uniformly charged ring is a classic result that combines symmetry arguments with integration.

Consider a thin ring of radius R carrying total charge Q , uniformly distributed. The field at a point P on the axis at distance x from the center is:

E = 1 4 _0 Qx (x^2 + R^2)^ 3/2

Direction: along the axis (away from the ring for positive Q). By symmetry, all perpendicular components cancel — every element dq has a diametrically opposite element whose perpendicular component is equal and opposite.

• At the center of the ring ( x = 0 ): E = 0 (all components cancel by symmetry)
• At x R : E kQ x^2 (ring looks like a point charge from far away)
• Maximum field occurs at x = R 2 (found by differentiating E w.r.t. x and setting dE/dx = 0 )
• This result is the building block for deriving the field of a disc and other shapes

Quick recall

At what distance from the center of a charged ring of radius R is the electric field maximum?

x = R/ 2 0.707R . At this point, E_ = Q 6 3 _0 R^2 . This is a very popular NEET question — memorize x = R/ 2 .

The ring formula is also the starting point for deriving the field of a uniformly charged disc (integrate rings of increasing radius from 0 to R) — a common board exam derivation.

Electric Flux

Electric flux measures 'how much electric field passes through a surface.' It's the dot product of the field with the area vector, and it's the key concept needed to understand Gauss's Law.

_E = E A = EA

Electric Flux

Electric flux quantifies how much electric field 'pierces through' a given surface. Though nothing actually flows, the concept is borrowed from fluid flow.

Electric flux ( _E ) through a surface is defined as the dot product of the electric field E with the area vector A (whose direction is the outward normal to the surface). For a uniform field and a flat surface:

_E = E A = EA

Here is the angle between E and the outward normal n to the surface. For a non-uniform field or a curved surface, we integrate:

_E = E d A

• SI unit of flux: N·m²/C or equivalently V·m (volt-meter)
• = 0° (field perpendicular to surface): = EA (maximum)
• = 90° (field parallel to surface): = 0 (field skims the surface, nothing passes through)
• = 180° (field enters the surface): = -EA (negative flux = field entering)
• Flux is a scalar quantity (the dot product gives a number, not a vector)

Flux Through Different Orientations

The angle between the field and the surface normal determines whether flux is positive, negative, or zero. Mastering this is crucial for Gauss's law applications.

The sign and magnitude of flux depend entirely on the angle between E and the outward normal n . Here's how different orientations affect flux:

• = 0° : Field lines enter perpendicularly through the surface → = +EA (max positive flux)
• 0° < < 90° : Field lines enter at an angle → = EA (positive, reduced)
• = 90° : Field lines are tangent to surface → = 0 (no flux at all)
• 90° < < 180° : Field lines exit at an angle → is negative
• = 180° : Field lines exit perpendicularly → = -EA (max negative flux)

Quick recall

A uniform field E passes through a cube. What is the net flux through the cube if it contains no charge?

Zero. Every field line that enters one face exits through another. The positive flux through exit faces exactly cancels the negative flux through entry faces. This is consistent with Gauss's law: = q_ enc / _0 = 0 .

Gauss's Law & Applications

Gauss's Law connects the electric flux through any closed surface to the charge enclosed within it. When combined with symmetry, it becomes the most powerful tool for finding electric fields of symmetric charge distributions.

E d A = q_ enc _0

Gauss's Law

The total electric flux through any closed surface equals 1/ε₀ times the total charge enclosed by that surface. Charges outside contribute zero net flux.

Gauss's Law states: the total electric flux through any closed surface (called a Gaussian surface) is equal to 1/ _0 times the net charge enclosed within that surface.

E d A = q_ enc _0

The beauty of this law is that it doesn't matter what shape the Gaussian surface is, or where inside the surface the charges are. Only the total enclosed charge matters. Charges outside the surface create field lines that enter and exit, contributing zero net flux.

• Gauss's Law is always true — it's a fundamental law of electromagnetism
• But it's useful for calculation only when there's enough symmetry to pull E out of the integral
• The Gaussian surface is an imaginary mathematical surface — it doesn't have to coincide with any physical surface
• Charges outside the Gaussian surface contribute zero NET flux (their entry flux = exit flux)
• q_ enc is the algebraic sum of charges inside — a +5 C and -3 C inside gives q_ enc = +2 C

Choosing the Right Gaussian Surface

The power of Gauss's law lies in choosing a Gaussian surface that exploits symmetry, making E constant on the surface and simplifying the integral.

Gauss's law is mathematically exact but practically useful only when we can simplify E d A into E A . This requires choosing a Gaussian surface where E is constant and either perpendicular or parallel to the surface everywhere.

Gauss's Law problem-solving strategy

1. Identify the symmetry of the charge distribution (spherical, cylindrical, or planar)
2. Choose a Gaussian surface matching that symmetry
3. Ensure E is either perpendicular to or parallel to every part of the Gaussian surface
4. On surfaces where E d A : the flux through that part is zero
5. On surfaces where E d A : pull E out of the integral → = E A
6. Find q_ enc inside the surface and apply EA = q_ enc / _0
7. Solve for E

Application: Infinite Line Charge

Using a cylindrical Gaussian surface around an infinite line charge to derive the electric field that falls as 1/r.

Consider an infinitely long straight wire with uniform linear charge density (C/m). By symmetry, E points radially outward (for > 0 ) and depends only on the perpendicular distance r .

Derivation using cylindrical Gaussian surface

1. Choose a coaxial cylinder of radius r and length l as the Gaussian surface
2. Curved surface: E d A → flux = E 2 r l
3. Two flat caps: E d A → flux = 0 (field is radial, caps are perpendicular to the wire)
4. Total flux: = E 2 r l
5. Enclosed charge: q_ enc = l
6. Gauss's law: E 2 r l = l / _0
7. Solve: E = 2 _0 r

E = 2 _0 r = 2k r

• Field falls as 1/r (NOT 1/r^2 ) — slower than a point charge
• Direction: radially outward for > 0 , inward for < 0
• The result is independent of the length l of the Gaussian surface — as expected for an infinite wire
• Valid approximation for a finite wire at points close to the wire and far from the ends

Application: Infinite Plane Sheet

An infinite uniformly charged plane creates a constant electric field that is the same at all distances — a remarkable result derived from Gauss's law.

Consider an infinite plane sheet with uniform surface charge density (C/m²). By symmetry, E is perpendicular to the sheet and has the same magnitude on both sides.

Derivation using pill-box Gaussian surface

1. Choose a cylindrical 'pill-box' of cross-section area A straddling the sheet — half above, half below
2. Two flat faces: E d A → flux through each face = EA
3. Curved surface: E d A → flux = 0
4. Total flux: = 2EA
5. Enclosed charge: q_ enc = A
6. Gauss's law: 2EA = A / _0
7. Solve: E = 2 _0

E = 2 _0 (single infinite sheet)

• Field is uniform — constant magnitude at all distances
• Direction: away from sheet (both sides) for > 0
• For TWO parallel sheets with + and - : E = / _0 between them, E = 0 outside (parallel plate capacitor!)
• For TWO parallel sheets with same sign ( + each): E = 0 between them, E = / _0 outside

Quick recall

Two infinite parallel plates carry charges + and - . What is the field (a) between the plates and (b) outside?

(a) Between: fields from both sheets point in the same direction → E = /(2 _0) + /(2 _0) = / _0 . (b) Outside: fields point in opposite directions → E = 0 . This is the basis of the parallel plate capacitor.

Application: Uniformly Charged Sphere

The electric field of a uniformly charged spherical shell and solid sphere, derived using Gauss's law. The 'shell theorem' for charges is identical to Newton's for gravity.

Spherical shell (charge Q on a thin shell of radius R ): choose a concentric spherical Gaussian surface of radius r .

E = cases 1 4 _0 Q r^2 & (r > R) 0 & (r < R) cases

Outside ( r > R ): the shell behaves exactly like a point charge Q at its center. Inside ( r < R ): the field is exactly zero — no enclosed charge. This is the electrostatic shell theorem, analogous to Newton's shell theorem for gravity.

Solid sphere (charge Q uniformly distributed throughout volume of radius R , with = 3Q 4 R^3 ):

E = cases 1 4 _0 Q r^2 & (r > R) 1 4 _0 Qr R^3 & (r < R) cases

• Shell — outside: acts as point charge at center ( E 1/r^2 )
• Shell — inside: E = 0 everywhere (this is why a Faraday cage works!)
• Solid sphere — outside: same as shell ( E 1/r^2 )
• Solid sphere — inside: E r (field increases linearly from center to surface)
• At the surface ( r = R ): both give E = kQ/R^2 — the field is continuous

Quick recall

A conducting sphere of radius R carries charge Q . What is the field at r = R/2 ?

Zero. A conducting sphere has all charge on its surface (it's a shell, not a solid sphere). Inside a conducting shell, E = 0 . Don't confuse conducting sphere (charge on surface only) with uniformly charged solid sphere (charge throughout volume).