Charges Conductors & Coulomb's Law Charges, Conductors and Coulomb's Law Electric charge is a basic property of matter, like mass, that causes electrical effects. Rubbing a plastic scale on dry hair makes tiny bits of paper jump; a thundercloud lights the sky; a CRT television attracts dust to its screen. These everyday events come from charge being transferred, separated or moved. Charge comes in two types, called positive and negative by convention. Like charges repel; unlike charges attract. The strength of this interaction is enormous compared to gravity at the scale of atoms, which is why electrons stay in atoms and why static shocks feel sharp. At school scale, we treat charges as point-like when their size is negligible compared to the distance between them. That lets us write laws using simple distances and vectors. In metals, charges that can move (free electrons) flow easily; in insulators like plastic or glass, charges are bound and cannot roam freely. When a conductor is connected to the Earth (ground), excess charge can leave or arrive until its potential matches Earth, neutralising it. When an insulator is brought near a charged object, its molecules shift slightly so that one side becomes a bit more positive and the other more negative; this is polarization and explains why a charged comb attracts neutral paper bits. Three principles guide all problems: charge is quantized in units of the elementary charge e = 1.602 10 -19 , C , charge is conserved in all processes (it can be moved or separated but the algebraic sum in an isolated system stays fixed), and charges add up algebraically (with signs). To calculate how strongly two charges pull or push, we use Coulomb's law. It says the force magnitude is proportional to the product of the charges and inversely proportional to the square of the distance between them. The direction is along the line joining the charges: repulsion for like signs, attraction for unlike. When more than two charges act, forces add as vectors (superposition). This chapter also links force to the idea of electric field E (force per unit positive charge at a point), introduces electric potential (energy per unit charge), and previews capacitance (how much charge you can store per volt). We will keep assumptions clear: electrostatics means charges are at rest, we use SI units, and we use the Cartesian sign convention for distances and directions. Big picture: Charge is the "source". Force tells you how two charges interact. Field is force per unit charge everywhere in space. Potential is energy per unit charge. Capacitance tells how much charge a conductor can store for a given potential difference. remember Electric charge (q) Intrinsic property of matter that causes electrical interactions. Comes in two types: positive and negative. SI unit is coulomb ( C ). Any free, isolated charge is an integer multiple of the elementary charge: q = n e where n Z and e = 1.602 10 -19 , C . Quantization of charge In an isolated system, algebraic sum of charges remains constant during any physical or chemical process. Conservation of charge Total charge of a system is the algebraic sum of individual charges, considering their signs. Charge is a scalar for addition. Additivity of charge Material with plenty of free charges (usually electrons) that can move through the bulk. In electrostatic equilibrium, E inside is zero and excess charge resides on the surface. Conductor Material in which charges are bound to atoms or molecules. No free charge motion through the bulk, but polarization can occur in an external field. Insulator (dielectric) Connecting a conductor to Earth so that charge flows until the conductor’s potential becomes equal to Earth’s (taken as zero). Grounding (Earthing) Slight displacement of bound positive and negative charges within atoms or molecules under an external electric field, producing induced dipoles. Polarization Charge quantization Any isolated charge is an integer multiple of the elementary charge. This principle shows that all isolated electric charges observed in nature must be integer multiples of the elementary charge 'e'. Quantization is powerful in estimates: 1 , C means about 6.25 10 18 electrons worth of negative charge. Typical static charges we handle are in microcoulombs ( C ) or nanocoulombs ( nC ). Conservation means if you rub a balloon on wool, electrons move from one to the other; total algebraic charge of the pair stays the same. Additivity lets us track charge transfer in steps: if a neutral body touches a charged conductor and then is removed, you can compute final charges by sharing rules and the total remaining constant. tip Coulomb’s law applies to point charges (or spherically symmetric charge distributions when you are outside them), at rest, in vacuum/air. At very small separations (comparable to atomic sizes) or in materials with polarization, local fields and quantum effects can modify the simple 1/r 2 form. Coulomb's law: magnitude of the force between two point charges separated by distance r. F = k e |q 1 q 2| r 2 Charges are stationary (electrostatics). Charges are point-like compared to separation r. Medium is vacuum (or air) so k e = 1/(4 0) . Electrostatic force magnitude between two point charges at rest, inverse-square in separation. F 12 is the force on charge 1 due to 2, directed along the line from 1 to 2 (repulsive if q 1 q 2>0 , attractive if q 1 q 2<0 ). Vector form Superposition means forces from multiple charges add like ordinary vectors. Compute each pairwise force using Coulomb’s law, assign correct direction along the line joining the source and the test charge, and then add geometrically. Symmetry can save time: on the perpendicular bisector of two equal like charges, forces are equal in magnitude and horizontal components cancel while vertical components add; at the midpoint of +q and −q, forces add in the same direction. Net force is the vector sum of individual forces due to each source charge. Superposition of forces Use F = k e |q 1 q 2|/r 2 . Sign product determines attraction/repulsion; magnitude uses absolute value. easy Two point charges q 1 = +2 , C and q 2 = -3 , C are placed 0.20 , m apart in air. Find the magnitude and nature (attractive/repulsive) of the force on each. q1 = +2 µC = +2 10 -6 C q2 = -3 µC = -3 10 -6 C r = 0.20 m k e = 8.9875 10 9 N·m 2 /C 2 Magnitude F and whether the force is attractive or repulsive. Always convert: C = 10 -6 , C , nC = 10 -9 , C , distance in metres. A common slip is to forget squaring r in F 1/r 2 . neet-alert q = 2 10 -6 C a = 0.10 m k e = 8.9875 10 9 N·m 2 /C 2 Each neighbor exerts repulsive force of magnitude F 0 = k e q 2/a 2 at 60 to each other. Resultant is along the angle bisector. Three identical charges q = +2 , C are placed at the corners of an equilateral triangle of side a = 0.10 , m . Find the net force on one charge due to the other two. medium Magnitude and direction of net force on one corner charge. How do materials respond to charge? In conductors, excess charge migrates until the whole conductor reaches a single potential. The electric field inside then becomes zero; otherwise free charges would continue to move. Any excess resides on the outer surface, crowding more at sharp points where curvature is high (hence lightning rods). In insulators, charges cannot travel through the bulk, but molecules rotate or stretch slightly to align with an external field, producing induced surface charges that can attract nearby objects without direct contact. Semiconductors sit in between; their conductivity depends on doping and temperature. Property Conductor Insulator Examples Free charge availability Abundant (electrons) None (bound charges) Copper, aluminium vs. glass, plastic Electric field inside (electrostatic equilibrium) Zero Non-zero possible Charge location (excess) Surface only Distributed locally where placed Response to external field Redistribution of free charges Polarization (induced dipoles) Grounding effect Neutralises quickly Little effect Charging methods: by friction (rub two different materials so that electrons transfer), by conduction (touch a charged conductor to a neutral conductor so some charge flows), and by induction (use a nearby charged body to shift charges in a conductor, then ground to remove one kind of charge before disconnecting). Induction is non-contact: it relies on the mobility of charges in a conductor and results in a net charge even though the inducing charge never touches. In insulators, charging is typically by rubbing, while induction mainly causes polarization rather than a lasting net charge. Bring a negatively charged rod near the neutral metal sphere without touching. Free electrons in the sphere are repelled away, leaving the near side positively charged and far side negatively charged. Connect the far side of the sphere briefly to ground. Electrons flow to Earth, leaving an excess of positive charge on the sphere. Remove the grounding connection first, then move the charged rod away. The sphere remains with a net positive charge. Charging a metal sphere by induction (steps) remember Electrostatic equilibrium in a conductor: net electric field inside is zero; potential is constant throughout; any excess charge resides on the outer surface; field at the surface is perpendicular to it. E = F q Test charge is positive and small enough not to disturb the source charges. Electrostatic conditions prevail. Definition of electric field intensity: E = F/q Electric field is the force per unit positive test charge at a point. Field lines visualize E : they start on positive charges and end on negative charges, never cross, and their density indicates field strength. A positive test charge placed at a point feels force tangent to the field line there. Remember, field lines are a drawing aid; the true field is a vector at every point in space. Superposition applies to fields: E net = E i . That is why symmetry arguments and vector addition are your best tools in multi-charge problems. Electric flux through a flat surface in a uniform field: E = E A E = E A Uniform electric field. Flat surface of area A. Flux measures how much field passes through a surface; use the angle with the normal. Gauss’s law (studied in detail later) says the net electric flux through a closed surface equals the enclosed charge divided by 0 . Symmetry lets us pick clever Gaussian surfaces where E is constant on the surface and the dot product is simple. That is how we get quick results for fields of a spherical shell, infinite line charge, and infinite plane sheet—without summing Coulomb forces piece by piece. These results also check our intuition: 1/r 2 outside a sphere (like a point charge), 1/r near a long line, and constant field near a broad plane. Field inside a uniformly charged thin spherical shell is zero; outside it is as if all charge were at the center. Electric field of a uniformly charged thin spherical shell Thin spherical shell of radius R with total charge Q uniformly spread. Electrostatics; vacuum. E = cases 0 & r < R 1 4 0 Q r 2 & r R cases Field at distance r from a uniformly charged infinite line: E = /(2 0 r) , radial direction. Infinite straight wire with linear charge density . Vacuum; electrostatics. E = 2 0 r Electric field due to an infinitely long uniformly charged wire Field near a uniformly charged infinite non-conducting sheet is constant: E = /(2 0) . Electric field due to an infinite plane sheet of charge Uniformly charged infinite non-conducting sheet with surface density . Vacuum; electrostatics. E = 2 0 Electric field vs distance for a uniformly charged thin spherical shell. control dependent custom For a uniformly charged thin spherical shell: E = 0 for r < R; at r = R, E jumps to Q/(4πε0 R 2 ); for r > R, E falls off as 1/ r 2 . Distance r from center Center: E = 0 At surface Q/(4πε0 R 2 ) At r = 2R Q/(16πε0 R 2 ) 2R Electric field E N/C Electric potential V is energy per unit charge. It is a scalar, so it adds algebraically without directions. Where field requires vector addition, potential lets you use simple sums. Work-energy viewpoint: the work done by an external agent in moving a test charge slowly from infinity to a point against the electric field increases its potential energy. Dividing by the charge gives potential, which depends on the source charges and the geometry. Zero potential is a reference; we usually choose V = 0 at r for isolated charges, but only potential differences are physically essential. Definition of electric potential: V = U/q Quasi-static movement of test charge so no radiation or kinetic energy change. Reference potential chosen (usually zero at infinity). V = U q Potential is energy per unit charge at a point; a scalar that adds directly. V = k q r Point charge q at rest in vacuum. Reference V( ) = 0. Potential due to a point charge Scalar potential falls as 1/r; sign follows the sign of the source charge. U = k q 1 q 2 r Charges fixed at separation r in vacuum. Reference U( ) = 0. Potential energy of two point charges Energy stored by a pair of charges: positive for like charges, negative for unlike. Electric dipole A pair of equal and opposite charges separated by a small distance. Dipole moment p = q , d , directed from negative to positive charge. p = q d Two point charges +q and −q separated by vector d . Electric dipole moment Vector measure of a dipole’s strength and orientation: p = q , d (from − to +). E = 1 4 0 p r 3 1 + 3 2 Short dipole (length 2a) so that r ≫ a (far-field approximation). Vacuum; electrostatics. Electric field of a short dipole at a distant point Far field of a short dipole falls as 1/ r 3 ; axial field is twice the equatorial field. Capacitance is a preview here because it links charge and potential: a conductor’s ability to store charge per unit potential is C = Q/V . For a given geometry and medium, C is fixed; if you connect the conductor to a battery that holds V constant, the charge on it becomes Q = CV . Parallel plate capacitors are the simplest model: two large plates facing each other. Inserting a dielectric between them increases capacitance because the dielectric polarizes, reducing the effective field and thus the potential for a given charge. Although we study capacitors in detail later, the definition and the common results appear often in mixed problems. Capacitance is the ratio of charge stored to potential difference: C = Q/V . C = Q V Two conductors at static equilibrium with charges +Q and −Q. Potential difference V between them is well-defined. Definition of capacitance For large plates separated by distance d (ignoring fringing): C = K 0 A/d . Capacitance of a parallel plate capacitor Plate area A is much larger than separation d (edge effects negligible). Uniform dielectric of relative permittivity K fills the space. C = K 0 A d With a dielectric slab of thickness t inserted: C = 0 A/(d - t + t/K) . Capacitance with a dielectric slab of thickness t C = 0 A d - t + t/K Dielectric slab of thickness t covers full plate area; rest is air/vacuum. Fields are uniform in each region; edge effects ignored. Distance x from the smaller charge +Q (at x=0) to the zero-force point. Q (symbolic) Positions: 0 and L = 0.30 m Test charge is positive hard Two like charges +Q and +4Q are fixed on a line at x=0 and x=L=0.30 , m . Where on the line between them should a positive test charge be placed so that the net electrostatic force on it is zero? Fields due to like charges oppose between them. Set magnitudes equal: kQ/x 2 = k(4Q)/(L - x) 2 . neet-alert When using the vector form, do not plug signed charges into the magnitude formula. Use |q 1 q 2| for magnitude, decide direction separately from the sign. Mixing signs inside the square root or forgetting the unit vector is a classic error. The smaller charge feels a smaller force than the larger one. By Newton’s third law, interaction forces are equal in magnitude and opposite in direction for the pair, regardless of their magnitudes. Accelerations differ because masses differ, not forces. If the electric field at a point is zero, the potential must be zero too. Field zero means potential is locally flat (no slope), not necessarily zero. Inside a charged spherical shell, E=0 but V is constant and generally non-zero. “Plus to minus for force on +q; minus to plus for dipole p.” Translation: A positive test charge feels force along field lines (from + to −), while the dipole moment p points from negative to positive charge. Units and quick conversions 1 , C = 10 -6 , C , 1 , nC = 10 -9 , C , 1 , pC = 10 -12 , C k e = 1/(4 0) 9.0 10 9 , N ,m 2/C 2 Field unit: N/C (equivalently V/m ). Potential unit: V = J/C Surface charge density : C/m 2 ; linear density : C/m Conversion discipline wins marks. Keep a tidy chain: write given values with SI units, show the power of ten, substitute into the formula, keep two significant figures for final NEET answers unless options demand otherwise. Check direction separately from magnitude. For symmetry cases, sketch the geometry and mark equal magnitudes and angles before summing vectors; that will often reduce the algebra to a single component. Finally, do an order-of-magnitude check: if you doubled r , did your computed F shrink by about four times? Elementary charge Smallest free charge magnitude e = 1.602 10 -19 , C (carried by proton or electron). k e = 1/(4 0) 8.9875 10 9 , N ,m 2/C 2 Coulomb’s constant Electrostatic constant k e Force per unit positive charge at a point: E = F /q Electric field Measure of field lines through a surface: E = EA Electric flux Energy per unit charge: V = U/q Potential Vector p = q , d from − to + charge. Dipole moment Charge stored per volt: C = Q/V Capacitance Charge per unit area on a surface: Surface charge density Key terms recap