Ohm's Law & Resistivity Think of electrical current not as a lightning bolt, but as a crowd of people (electrons) trying to walk through a busy market street (the wire). The 'Voltage' is the urgency or push—like a siren scaring them to run faster. The 'Resistance' is the density of the crowd and stalls—the collisions they make with people and objects standing still (ions). Ohm's Law ( V=IR ) essentially says: The speed of the crowd ( I ) depends on how hard you push them ( V ) divided by how difficult the path is ( R ). For NEET, remember: Resistance is the toll tax the wire charges for the current to pass, and that tax is paid in the form of Heat. Ohm's Law & Resistivity Electric current in a metal is the slow drift of a gigantic number of electrons through a lattice of fixed positive ions. When a battery is connected, it sets up an electric field in the wire. Each electron gets accelerated but keeps colliding with ions; the average effect is a small steady drift velocity opposite to the field. Because the lattice and temperature are the same throughout a uniform wire, the ease or difficulty of this drift remains fixed, so the current through the wire is directly proportional to the applied potential difference. This linear rule is Ohm’s law. It works beautifully for metallic conductors under fixed physical conditions (chiefly constant temperature and no mechanical strain). A material’s internal opposition is summarized by its resistivity , which depends only on the substance and temperature, not on the piece’s size. A long thin piece of the same metal has a larger resistance R than a short thick piece, but both share the same . In problems, you will often convert between the geometry-based resistance R= , l A and the circuit-level Ohm’s law V=IR . Graphs of V vs I help you see whether a device is ohmic (straight line through origin) or non-ohmic (curved or passing away from origin, like diodes and filament lamps). Temperature changes are crucial: for metals, higher temperature increases ion vibrations and collisions (resistance rises); for semiconductors/electrolytes, extra charge carriers dominate (resistance falls). Keeping these pictures in mind lets you quickly judge the direction of change before calculating. remember Water-tank analogy: Water height ≈ voltage (push), pipe narrowness ≈ resistance (opposition), water flow rate ≈ electric current. Raising the tank height increases flow; squeezing the pipe reduces flow for the same height. Voltage (Potential Difference) Work done per unit charge to move charge between two points. Drives the current. SI unit: volt ( V ). Current ( I ) Rate of flow of charge through a cross-section. I= dq dt . SI unit: ampere ( A ). Opposition offered by a conductor to current. R= V I . SI unit: ohm ( ). Resistance ( R ) Resistivity ( ) Material property that sets resistance for a given geometry. R= , l A . Unit: , m . Conductivity ( ) Ease of conduction per material. = 1 . Unit: S ,m -1 . Current per unit area in vector form. Linked to field by J= E for ohmic media. Current Density ( J ) Average net velocity of electrons due to the field, opposite to E . Related by I=n e A v d . Drift Velocity ( v d ) Number Density ( n ) Free electrons per unit volume of the conductor. Relaxation Time ( ) Average time between successive collisions of electrons with the lattice. Temperature Coefficient ( ) Fractional change in resistance per degree at a reference temperature: R t=R 0[1+ (t-t 0)] . Ohmic devices obey V I (straight-line V–I). Non-ohmic devices have nonlinear V–I (e.g., diode, filament lamp). Ohmic vs Non-ohmic Magnitude of drift velocity per unit electric field: v d= E . For metals, = e m . Mobility ( ) Ohm’s Law: Statement and Equation Ohm's law (macroscopic) At fixed physical conditions (temperature, strain, etc.), current is directly proportional to applied potential difference. Applicability: Metallic conductors at constant temperature, no mechanical strain, and low to moderate fields. Fails for semiconductors, electrolytes, vacuum tubes, and metals at high temperature where R changes significantly. tip Slope of V–I graph for an ohmic conductor equals resistance; reciprocal slope of I–V graph equals resistance. Definition of resistance Geometry relation Resistance depends on length l and cross-sectional area A of a uniform wire made of a material with resistivity . Applies to uniform conductors where the current density is assumed to be constant across the cross-section. Metal has n free electrons per unit volume. Uniform wire of length l and cross-section A . Steady state with average relaxation time . Electric field E= V l inside the wire. R = ( m ne 2 ) l A Drift velocity from equation of motion with frequent collisions. Current in terms of E . Relating field to potential difference. Identify R=V/I . Microscopic origin of Ohm’s law and R= ( m ne 2 ) l A The factor m ne 2 is purely material-dependent and is defined as the resistivity . Increasing n (more carriers) or (fewer collisions) reduces ; increasing the electron mass m would increase . Geometry only scales the total resistance through l and A . Conductivity grows with carrier density and relaxation time. Resistivity and conductivity This model applies to the electrical conduction in metals and semiconductors, assuming the free electron gas model and the relaxation time a Vector form Material law: current density is proportional to electric field in an ohmic medium. Describes the relationship between current density and electric field strength in a conductive medium, particularly useful when current flow Be careful with geometry transformations. When a wire is stretched without changing volume, the length increases while area decreases. Since R= , l A and A 1 l for constant volume, resistance changes as the square of the stretch factor. If a wire is stretched to n times its length at constant volume, the new resistance is R'=n 2 R . Many students wrongly use R' l only and miss the area change. neet-alert Temperature Dependence Valid for moderate temperature intervals. For metals >0 ; for semiconductors/electrolytes <0 . Linear law (small range) This formula provides a linear approximation for the change in resistance when the temperature variation is small, making it useful for mode Use temperature difference in either °C or K—numerically the same change. carries units K -1 (or C -1 ). tip Why do metals show >0 ? As temperature rises, lattice ions vibrate more strongly, decreasing and hence conductivity ( ) falls. In intrinsic semiconductors, the number of charge carriers n increases very rapidly with temperature and dominates over the slight decrease of , so increases and R decreases. Resistivity depends on the shape or size of the sample. Resistivity is an intrinsic property depending only on material and temperature. Resistance R depends on l and A ; does not. Ohm’s law holds for every conductor at all currents. It holds for ohmic materials under fixed conditions. Diodes, filament bulbs, electrolytes, and semiconductors are non-ohmic; metals at high temperature or very high fields also deviate. Ohmic and Non-ohmic V–I Characteristics For an ohmic conductor, the V–I plot is a straight line through the origin with constant slope V I =R . In a filament lamp, heating raises R as current grows, giving a curve that flattens (slope increases) with I . A p–n diode conducts significantly only in forward bias beyond threshold; its V–I is highly nonlinear and does not pass through the origin. Straight line through origin for an ohmic resistor; upward-bending curve for a filament lamp indicating rising resistance with temperature. iv A filament lamp instead bends upward as resistance rises with temperature. Voltage V Current I Ohmic line derived Filament curve derived R I Slope = R 2D PLOT Ohmic conductor: V vs I V = R I Resistance Hydraulic analogy vs electrical circuit: water height as voltage, pipe narrowness as resistance, and flow rate as current in a battery–resistor loop. Water tank and pipe analogy compared with a battery–resistor circuit side-by-side. Lattice with electrons showing drift opposite to electric field. Microscopic view: orange lattice ions and blue electrons. Electric field to the right; electrons drift slowly to the left with random collisions. A multimeter near a sloped plane representing resistance on a V–I graph. 3D-style V–I plot: the bright slope labelled Resistance with a digital multimeter indicating example readings. How to read the slope correctly: In a V–I graph (V on y-axis), slope equals V I =R . In an I–V graph (I on y-axis), the slope is I V = 1 R . Many errors on NEET stem from mixing these up. Connect a variable DC source, an ammeter in series, and a voltmeter across the resistor. Change the source voltage in small steps; record V and I after waiting for steady values. Plot V on y-axis and I on x-axis; the straight line through origin confirms ohmic behaviour. Slope =R . Check that R is the same for increasing and decreasing runs (temperature stability). Experiment sketch: Obtain V–I of a resistor Why resistivity matters Designing cables: choose low- metals (Cu, Al) to reduce I 2 R losses. Heating elements: choose alloys with high and stable (nichrome, manganin). Sensors: semiconductors where resistivity changes sharply with temperature or doping. Safety: correct wire thickness from R= l/A to avoid overheating. Material Type Approx. Resistivity at 20°C (Ω·m) Notes Representative NCERT-order magnitudes; actual values vary with purity and temperature. Silver Metal 1.6×10 -8 Lowest among metals; expensive. Copper Metal 1.7×10 -8 Standard for wiring; >0 . Aluminium Metal 2.6×10 -8 Lightweight; used in transmission lines. Iron Metal 1.0×10 -7 Higher than Cu/Al. Nichrome Alloy 1.1×10 -6 Stable with temperature; heating coils. Graphite (carbon) Semimetal 3.5×10 -5 Anisotropic conduction. Germanium Semiconductor 0.46 Resistivity falls with T ; <0 . Silicon (intrinsic) Semiconductor 2.3×10 3 Strongly temperature dependent. Glass Insulator 10 10 –10 14 Large range; depends on composition. Hard rubber Insulator 10 13 –10 16 Very poor conductor. Orders of magnitude separate conductors, semiconductors, and insulators. Expect metals at 10 -8 – 10 -6 , , m , semiconductors around 10 -3 to 10 4 , and insulators above about 10 8 . When a question gives a material, you can often anticipate whether I will be large or tiny before computing. Rho–L–A: “Rho times Length over Area” — R= , l A . If l doubles and A halves, R becomes 4 times. neet-alert Graph trap: In a V–I plot, slope = R . In an I–V plot, slope = 1/R . Many MCQs silently swap axes. Current is the time rate of flow of charge: I= dq dt . Distinguish from average current I avg = q t . Charge crossing a surface can vary with time. Define average current over a short interval. I = dq dt Instantaneous current I= dq dt Average rate of charge flow in time t . Take the limit of the average as the interval shrinks. Large current can arise from small v d if n and A are large. Microscopic current Typical electron drift speeds are millimetres per second, not anywhere near the speed of light. The electrical effect reaches quickly because the electric field throughout the conductor is established almost instantly (at a significant fraction of light speed), while individual electrons drift slowly. Electrons in a wire move nearly at the speed of light when current flows. The drift speed v d is tiny (mm/s). It is the electric field that propagates rapidly, not the carriers themselves. Power and heating are tied to Ohm’s law: P=VI=I 2 R= V 2 R . In wire-sizing questions, balancing acceptable voltage drop and heat dissipation is essential; lower R reduces losses but needs larger A (more cost and weight). Ω, W V = 5.0 , V I = 0.50 , A Use R= V I and P=VI . easy R and P A resistor draws a current of 0.50 A when 5.0 V is applied. Find its resistance and the power dissipated. A copper wire has resistance R. It is drawn (stretched) to twice its original length with constant volume. Find the new resistance. R' Resistance scales as l 2 when volume is fixed. medium For constant volume, A' = A/2 . Use R= , l A . Initial resistance = R Final length = 2l (volume constant) hard Use R t=R 0[1+ (t-t 0)] and P= V 2 R at fixed V . Ω, % R 0 = 5.0 , at t 0=20 C = 0.004 , K -1 t = 120 C V = 10 , V A metal wire has resistance R 0=5.0 , at 20 C with temperature coefficient =0.004 , K -1 . It is carrying a current such that its temperature rises to 120 C . Find the new resistance and the percentage increase in power at a fixed applied voltage of 10 V. R t and percentage change in power at fixed voltage Notice how the temperature rise increased R , so at fixed V the current and power dropped. If instead the source were ideal current-controlled (fixed I ), power would increase as P=I 2 R with temperature. A cylindrical nichrome wire of length 2.0 m and cross-sectional area 0.50 , mm 2 is connected to 3.0 V. Take =1.1 10 -6 , , m . Find the current. l = 2.0 , m A = 0.50 , mm 2 = 0.50 10 -6 , m 2 = 1.1 10 -6 , , m V = 3.0 , V medium Compute R= , l A , then use I= V R . Superconductivity (qualitative): For some materials below a critical temperature T c , resistivity suddenly drops to effectively zero and magnetic fields are expelled (Meissner effect). This lies beyond the simple linear R – T law. remember Practical cautions: When measuring V – I , large currents heat the sample and spoil linearity, making R appear larger at higher readings. Use low currents or short duty cycles, or keep the sample thermally anchored. Edge conditions and limits As l 0 (a perfect short of the same material), R 0 ; as A 0 , R . At very high fields, electron energies and scattering change; the linear V I law can fail. At T 0 for normal metals, impurities dominate scattering so approaches a residual value (not zero), unless the material is a superconductor. Boundary condition for Eq. R t=R 0[1+ T] : use only over modest T where is nearly constant. neet-alert Unit vigilance: Resistivity in , m ; conductivity in S ,m -1 ; current density in A ,m -2 . Avoid mixing mm 2 with m 2 — convert carefully. Worked Reasoning: Connecting the Pieces Start from the material: pick (from a table or given). Convert geometry to R via R= ,l/A . If temperature differs from the reference, correct R using R t=R 0[1+ T] . Then enter the circuit world with V=IR and power relations. For graphs, identify which variable is on which axis before reading the slope. When non-ohmic, use local (differential) resistance r= dV dI at a point if asked. Microscopic parameters occasionally appear in questions: knowing I=neAv d and = ne 2 m lets you connect carrier density or mobility to macroscopic current. For example, doubling n halves R for the same geometry. Ohm’s law At constant physical conditions, V=IR . Resistance Opposition to current; R=V/I= l/A . Material property = m ne 2 ; independent of shape. Resistivity Conductivity =1/ = ne 2 m . Current (instantaneous) I= dq dt . Current density J= E . Temperature coefficient in R t=R 0[1+ (t-t 0)] . Key terms recap