Capacitance & Current Electric

Cheatsheet Content

### CONCEPT OF CAPACITANCE - **Definition:** Capacitance of a conductor is its ability to store charge. - **Formula:** $Q \propto V \Rightarrow Q = CV$ - **Unit:** Farad (F), Coulomb/Volt - **Dimension:** $[M^{-1}L^{-2}T^4A^2]$ - **Capacity:** Independent of charge, potential, material, and thickness. Practically limited by dielectric breakdown. ### THE CAPACITANCE OF A SPHERICAL CONDUCTOR - **Isolated Spherical Conductor:** $C = 4\pi\epsilon_0 R$ - **In a Medium:** $C_{medium} = 4\pi\epsilon_0\epsilon_r R$ - **Factors affecting Capacitance:** - Size and Shape of Conductor - Surrounding medium - Presence of other conductors nearby ### CONDENSER/CAPACITOR - **Definition:** A pair of conductors with opposite charges capable of accommodating sufficient charge. - **Principle of a Condenser:** Capacitance increases by reducing potential while keeping charge constant, often by inducing negative charge closer and earthing positive induced charges. ### ENERGY STORED IN A CHARGED CONDUCTOR/CAPACITOR - **Work Done (W) / Potential Energy (U):** - $W = \int V dq = \frac{Q^2}{2C}$ - $U = \frac{Q^2}{2C} = \frac{1}{2}CV^2 = \frac{1}{2}QV$ ### GOLDEN KEY POINTS (Capacitance) - **Earth's Capacity:** Assumed infinite ($C = \infty$). - **Actual Capacity of Earth:** $C = 4\pi\epsilon_0 R \approx 711 \mu F$. - **Work done by battery ($W_b$):** $W_b = QV$. - **Energy Stored:** $U = \frac{1}{2}QV$. 50% energy is lost as heat. ### REDISTRIBUTION OF CHARGES AND LOSS OF ENERGY - When connected, charge flows from higher to lower potential until potentials equalize. - **Common Potential (V):** $V = \frac{C_1V_1 + C_2V_2}{C_1 + C_2}$ - **Charges after connection:** $Q'_1 = C_1V$, $Q'_2 = C_2V$ - **Ratio of charges after redistribution (spherical conductors):** $\frac{Q'_1}{Q'_2} = \frac{C_1V}{C_2V} = \frac{R_1}{R_2}$ - **Loss of energy in redistribution ($\Delta U$):** $\Delta U = U_f - U_i = -\frac{1}{2} \frac{C_1C_2}{C_1+C_2}(V_1-V_2)^2$. Negative sign indicates energy decrease. ### PARALLEL PLATE CAPACITOR - **Capacitance (C):** - **Electric Field (E):** $E = \frac{\sigma}{\epsilon_0\epsilon_r}$ - **Potential Difference (V):** $V = Ed$ - **Capacitance:** $C = \frac{q}{V} = \frac{\epsilon_0\epsilon_r A}{d}$ - **Air/Vacuum:** $\epsilon_r = 1 \Rightarrow C_0 = \frac{\epsilon_0 A}{d}$ - **With Dielectric:** $C = \epsilon_r C_0 = KC_0$ (where $K$ is dielectric constant) - **Force between the plates:** - **Electric field due to positive plate:** $E = \frac{\sigma}{2\epsilon_0} = \frac{Q}{2\epsilon_0 A}$ - **Force on negative charge:** $F = -QE = -\frac{Q^2}{2\epsilon_0 A}$ - **Magnitude of force:** $F = \frac{Q^2}{2\epsilon_0 A} = \frac{1}{2}\epsilon_0 A E^2$ - **Force per unit area (Energy density/Electrostatic pressure):** $p = u = \frac{F}{A} = \frac{1}{2}\epsilon_0 E^2$ ### SPHERICAL CAPACITOR - **Outer sphere earthed:** - **Capacitance (in air/vacuum):** $C = 4\pi\epsilon_0 \frac{R_1R_2}{R_2-R_1}$ - **With medium:** $C = 4\pi\epsilon_0\epsilon_r \frac{R_1R_2}{R_2-R_1}$ - **Inner sphere earthed:** - System equivalent to spherical capacitor in parallel with spherical conductor of radius $R_2$. - **Total capacity:** $C = 4\pi\epsilon_0 \frac{R_1R_2}{R_2-R_1} + 4\pi\epsilon_0 R_2 = 4\pi\epsilon_0 \frac{R_2^2}{R_2-R_1}$ ### CYLINDRICAL CAPACITOR - **Capacitance (C):** - **Electric field between cylinders:** $E = \frac{\lambda}{2\pi\epsilon_0 r} = \frac{Q/l}{2\pi\epsilon_0 r}$ - **Potential difference:** $V = \int_{R_1}^{R_2} E dr = \frac{Q}{2\pi\epsilon_0 l} \ln\left(\frac{R_2}{R_1}\right)$ - **Capacitance (in air/vacuum):** $C = \frac{2\pi\epsilon_0 l}{\ln(R_2/R_1)}$ - **With medium:** $C = \frac{2\pi\epsilon_0\epsilon_r l}{\ln(R_2/R_1)}$ ### CAPACITY OF DIFFERENT CONFIGURATION - **Parallel Plate Capacitor:** $C = \frac{\epsilon_0 A}{d}$ - **Partially filled with dielectric:** $C = \frac{\epsilon_0 A}{d - t(1 - 1/\epsilon_r)}$ - **Fully present medium:** $C_{medium} = \frac{\epsilon_0\epsilon_r A}{d}$ - **Partially filled with conducting slab:** $C = \frac{\epsilon_0 A}{d-t}$ ### DISTANCE AND AREA DIVISION BY DIELECTRIC - **Distance Division:** - Area remains same, distance divided. - Capacitors are in series: $\frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2}$ - **Area Division:** - Distance remains same, area divided. - Capacitors are in parallel: $C = C_1 + C_2$ ### FORCE ON A DIELECTRIC IN A CAPACITOR - **Work done by external agent (W_e):** $W_e = -W_{electrostatic} = -\Delta U = F \cdot dx$ - **Force (F):** $F = \frac{Q^2}{2A\epsilon_0}\frac{d\epsilon_r}{dx}$ (for variable dielectric) - **With battery connected (constant potential difference V):** $F = \frac{1}{2} V^2 \frac{dC}{dx}$ ### CHARGING & DISCHARGING OF A CAPACITOR - **Charging:** - **Charge at any instant:** $Q = Q_0[1 - e^{-t/RC}]$ - **Current at any instant:** $i = i_0 e^{-t/RC}$ - **Potential at any instant:** $V = V_0(1 - e^{-t/RC})$ - **Time Constant ($\tau$):** $\tau = RC$. At $t=\tau$, $Q = 0.632 Q_0$. - **Discharging:** - **Charge at any instant:** $Q = Q_0 e^{-t/RC}$ - **Current at any instant:** $i = -i_0 e^{-t/RC}$ - **Potential at any instant:** $V = V_0 e^{-t/RC}$ - At $t=\tau$, $Q = 0.368 Q_0$. ### CURRENT ELECTRICITY - **Electric Current:** Flow of electric charges. - **Average Current:** $I_{av} = \frac{\Delta Q}{\Delta t}$ - **Instantaneous Current:** $I = \lim_{\Delta t \to 0} \frac{\Delta Q}{\Delta t} = \frac{dQ}{dt}$ ### GOLDEN KEY POINTS (Current) - **Current:** Fundamental quantity, dimension $[A]$, scalar quantity, SI unit Ampere. - **Ampere:** 1 Coulomb of charge flowing per second. - **Conventional direction:** Direction of positive charge flow. - Conductor remains uncharged as current flows. - Current is independent of cross-section. - **Current (n particles, charge q, area A):** $I = \frac{nqA}{t}$ - **Current (n particles/volume, charge q, velocity v, area A):** $I = nqvA$ - **Current (charge q, radius r, speed v):** $I = \frac{q}{T} = \frac{qv}{2\pi r}$ ### CLASSIFICATION OF MATERIALS ACCORDING TO CONDUCTIVITY - **Conductor:** Materials with free electrons (conduction electrons) that move in an electric field. - **Insulator:** Materials with tightly bound electrons, no free electrons. - **Semiconductor:** Materials with properties between conductors and insulators; conductivity increases with temperature. ### BEHAVIOR OF CONDUCTOR IN ABSENCE OF APPLIED POTENTIAL DIFFERENCE - Electrons have random thermal motion, zero average velocity, no net current. - **Thermal speed ($v_{rms}$):** $v_{rms} = \sqrt{\frac{3kT}{m}}$ (typically $10^5$ m/s at room temperature). - **Mean free path ($\lambda$):** Average distance between collisions ($\sim 10Å$). - **Relaxation time ($\tau$):** Average time between collisions ($\sim 10^{-14}$s). ### BEHAVIOR OF CONDUCTOR IN PRESENCE OF APPLIED POTENTIAL DIFFERENCE - Applied potential difference creates an electric field. - **Electric Field (E):** $E = \frac{V}{L}$ - **Acceleration of electron (a):** $a = \frac{F}{m} = \frac{-eE}{m}$ ### DRIFT VELOCITY - **Definition:** Velocity with which free electrons drift towards the positive terminal under an electric field. - **Formula:** $\vec{v}_d = a\vec{\tau} = \frac{e\vec{E}}{m}\tau$ (Order of $10^{-4}$ m/s). ### RELATION BETWEEN CURRENT AND DRIFT VELOCITY - **Number of free electrons in conductor of length L:** $nAL$ - **Total charge on these free electrons:** $\Delta q = neAL$ - **Time taken to cross conductor:** $\Delta t = \frac{L}{v_d}$ - **Current (I):** $I = \frac{\Delta q}{\Delta t} = neAv_d$ ### CURRENT DENSITY (J) - **Definition:** Vector quantity, magnitude is current per unit area normal to the direction of charge flow. - **Formula:** $\vec{J} = \frac{dI}{dA} \hat{n}$ or $dI = \vec{J} \cdot d\vec{A}$ - **S.I. unit:** Ampere/m$^2$ - **Dimension:** $[L^{-2}A]$ - **Direction:** Same as electric field $\vec{E}$. ### RELATION BETWEEN CURRENT DENSITY, CONDUCTIVITY AND ELECTRIC FIELD - **Drift velocity:** $v_d = \frac{eE}{m}\tau$ - **Current density:** $\vec{J} = ne\vec{v}_d = \frac{ne^2\tau}{m}\vec{E} = \sigma\vec{E}$ - **Conductivity ($\sigma$):** $\sigma = \frac{ne^2\tau}{m}$ (depends on material and temperature). - **Ohm's Law (Microscopic Level):** $\vec{J} = \sigma\vec{E}$ ### RELATION BETWEEN POTENTIAL DIFFERENCE AND CURRENT (Ohm's Law) - **Ohm's Law (Macroscopic Level):** $V = IR$ (if physical conditions remain same). - **Resistance (R):** Proportionality constant, opposition to current flow. - **Graph:** V-I graph for metallic conductor is a straight line. - **Slope:** Slope of V-I graph is resistance R. ### GOLDEN KEY POINTS (Current Density) - 1 Ampere = $6.25 \times 10^{18}$ electrons/second. - Current is scalar, current density is vector. - Free electron density in conductors $\sim 10^{28}$ electrons/m$^3$. - **Non-uniform cross-section:** I is same, J and $v_d$ depend on area. - **Temperature increase:** Decreases relaxation time, increases resistance. - **V-I curves:** Different at different temperatures. ### RESISTANCE - **Definition:** Opposition to the flow of charge due to collisions of electrons with ions. - **Unit:** Ohm ($\Omega$), Volt/Ampere. - **Dimension:** $[ML^2T^{-3}A^{-2}]$ - **Dependence:** - **Length (L):** $R \propto L$ - **Area (A):** $R \propto \frac{1}{A}$ - **Nature of material:** $R = \rho \frac{L}{A}$ - **Temperature:** $R_T = R_0(1 + \alpha\Delta T)$ - $\alpha > 0$ for metals. - $\alpha ### RESISTIVITY - **Definition:** Specific resistance of a material. - **Formula:** $\rho = \frac{RA}{L}$ - **Dependence:** - **Nature of material** - **Temperature** - **Independence:** Size and shape of the material. ### SPECIFIC USE OF CONDUCTING MATERIALS - **Microhm (heating elements):** High resistivity, high melting point. - **Tin-lead alloy (fuse wire):** Low melting point, low resistivity. - **Manganin/Constantan (resistance boxes):** Moderate resistivity, very small temperature coefficient. - **Tungsten (bulb filament):** Low resistivity, high melting point. - **Copper (connection wires):** Low resistance and resistivity. ### COLOUR CODE FOR CARBON RESISTORS - **Strips:** A (1st digit), B (2nd digit), C (multiplier), D (tolerance). - **Tolerance:** Gold ($\pm 5\%$), Silver ($\pm 10\%$), No color ($\pm 20\%$). - **Mnemonic:** BB ROY Great Britain Very Good Wife (Black, Brown, Red, Orange, Yellow, Green, Blue, Violet, Grey, White). ### COMBINATION OF RESISTORS - **Series Combination:** - **Current:** Same through each resistor. - **Voltage:** $V = V_1 + V_2 + V_3$ - **Equivalent Resistance:** $R_{eq} = R_1 + R_2 + R_3$ - **Parallel Combination:** - **Potential Difference:** Same across each resistor. - **Current:** $I = I_1 + I_2 + I_3$ - **Equivalent Resistance:** $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$ ### KIRCHHOFF'S LAW - **First Law (Junction Law or Current Law):** $\sum I = 0$ (Sum of currents entering a junction equals sum of currents leaving). Based on **Law of conservation of charge**. - **Second Law (Loop Rule or Potential Law):** $\sum E - \sum IR = 0$ (Algebraic sum of potential differences and EMFs in a closed loop is zero). Based on **Law of conservation of energy**. ### GOLDEN KEY POINTS (Resistors) - **Wire stretched n times length:** Resistance becomes $n^2$ times. - **Wire stretched, radius reduced 1/n:** Resistance becomes $n^4$ times. - **Maximum resistance:** Series connection. - **Minimum resistance:** Parallel connection. - **Ohm's Law:** Not fundamental. V-I relation can be non-linear, depend on sign of V, or be non-unique. - **Alloys:** Higher resistivity, lower temperature coefficient than pure metals. - **Non-metals (e.g., carbon):** Resistance decreases with increasing temperature. - **Insulators (e.g., amber):** Resistivity much greater than metals. - **Semiconductors (e.g., carbon, graphite):** Negative temperature coefficient. ### ELECTRO MOTIVE FORCE (E. M. F.) - **Definition:** Potential difference across cell terminals when no current is drawn. Energy per unit charge supplied by cell. - **Dependence:** Nature of electrolyte, metal of electrodes. - **Independence:** Area of plates, quantity of electrolyte, size of cell. ### TERMINAL VOLTAGE (V) - **Definition:** Potential difference across cell terminals when current is drawn or supplied. - **Discharging:** $V = E - Ir$ - **Charging:** $V = E + Ir$ - **Open Circuit:** $R = \infty \Rightarrow I = 0 \Rightarrow V = E$ - **Short Circuited:** $R = 0 \Rightarrow I = E/r \Rightarrow V = 0$ ### INTERNAL RESISTANCE - **Definition:** Opposition to current flow offered by the electrolyte inside the cell. - **Factors affecting internal resistance:** - **Distance between electrodes:** Increases with distance. - **Area dipped in electrolyte:** Decreases with area. - **Concentration of electrolyte:** Increases with concentration. - **Temperature:** Decreases with temperature. ### COMBINATION OF CELLS - **Series Combination:** - **Equivalent EMF:** $E_{eq} = E_1 + E_2 + ...$ - **Equivalent Internal Resistance:** $r_{eq} = r_1 + r_2 + ...$ - **Current (n identical cells):** $I = \frac{nE}{nr + R}$ - **Parallel Combination:** - **Equivalent EMF (m identical cells):** $E_{eq} = E$ - **Equivalent Internal Resistance (m identical cells):** $r_{eq} = \frac{r}{m}$ - **Current (m identical cells):** $I = \frac{mE}{mR + r}$ - **Mixed Combination (n cells in series, m such branches):** - **Total cells:** $nm$ - **Equivalent EMF:** $nE$ - **Equivalent Internal Resistance:** $\frac{nr}{m}$ - **Current:** $I = \frac{nE}{R + nr/m}$ - **Maximum current:** When $R = \frac{nr}{m}$ ### GOLDEN KEY POINTS (Cells) - **Charging:** $V = E + Ir$ - **Series combination useful:** When $nr R$. - **Maximum power in R:** When $R = r_{net}$. - **Ideal cell:** Internal resistance $r = 0$. - **Maximum current:** When external resistance $R = 0$. ### GALVANOMETER - **Definition:** Instrument measuring current strength by coil deflection due to magnetic field. ### SHUNT - **Definition:** Small resistance connected in parallel to a galvanometer coil. - **Merits:** Protects galvanometer, converts to ammeter, changes ammeter range. - **Demerits:** Decreases galvanometer sensitivity. ### CONVERSION OF GALVANOMETER INTO AMMETER - Connect low resistance (shunt $R_s$) in parallel. - **Shunt resistance:** $R_s = \frac{I_g R_g}{I - I_g}$ ### CONVERSION OF GALVANOMETER INTO VOLTMETER - Connect high resistance (R) in series. - **Series resistance:** $R = \frac{V}{I_g} - R_g$ ### GOLDEN KEY POINT (Galvanometer) - **Deflection variation:** Depends on magnitude of deflection. - **Suspended coil galvanometer:** Measures $\sim 10^{-9}$ Ampere. - **Ballistic galvanometer:** Measures charge flow for small time intervals. - **Ideal ammeter:** Zero resistance. - **Ideal voltmeter:** Infinite resistance. - **Bridge sensitivity:** Maximum when resistances in all four branches are of the same order. ### WHEAT STONE BRIDGE - **Balanced condition ($I_g = 0$):** $\frac{P}{Q} = \frac{R}{S} \Rightarrow PS = QR$ ### METRE BRIDGE - **Principle:** Wheatstone bridge. - **Used for:** Finding unknown resistance. - **Balance condition:** $\frac{R_B}{S} = \frac{l}{(100-l)}$ ### POST OFFICE BOX - **Principle:** Wheatstone bridge. - **Ratio arms:** Resistances between A-B and B-C. - **Known arm:** Resistances between A-D. - **Null deflection:** $\frac{Q}{P} = \frac{S}{R}$ - **Sensitivity:** Increases by decreasing Q/P ratio. ### POTENTIOMETER - **Necessity:** Measures potential difference without drawing current (infinite effective resistance). - **Working principle:** Balance unknown potential difference against a known, uniformly distributed potential difference along a wire (null deflection). - **Potentiometer wire:** Made of alloys (magnin, constantan, Eureka) with high specific resistance, negligible temperature coefficient. - **Potential gradient (x):** Fall of potential per unit length ($x = \frac{V}{L}$). ### CIRCUITS OF POTENTIOMETER - **Primary circuit:** Constant voltage source, rheostat. - **Secondary circuit:** Unknown EMF/galvanometer. - **Potential gradient:** Depends only on primary circuit. ### APPLICATIONS OF POTENTIOMETER - Measure potential difference. - Find EMF of a cell. - Compare EMFs ($E_1/E_2$). - Find internal resistance of a cell. - Compare resistances. - Find unknown resistance. - Find current in a circuit. - Calibrate ammeter/voltmeter. - Find thermocouple EMF. ### HEATING EFFECT OF CURRENT - **Cause of Heating:** Electrons accelerate, collide with ions, transfer energy as heat. - **Joule's Law of Heating:** $H = I^2Rt = Pt = VIt = \frac{V^2}{R}t$ - **Heat produced:** Independent of current direction. ### PRACTICAL UNITS - **Energy:** 1 kWh = $3.6 \times 10^6$ Joules, 1 BTU = 1055 J. - **Power:** Watt. - **Watt-hour meter:** Records electrical energy consumed. ### POWER TRANSFERRED TO LOAD BY CELL - **Power (P):** $P = I^2R = \frac{E^2R}{(r+R)^2}$ - **Maximum power:** Occurs when load resistance $R$ equals internal resistance $r$ ($R=r$). - **Maximum power formula:** $P_{max} = \frac{E^2}{4r}$ ### PARALLEL COMBINATION OF RESISTORS (Bulbs) - **Total power consumed:** $P_{total} = P_1 + P_2 + ...$ - **Brightness:** Brighter for bulbs with greater wattage (lower resistance). ### FUSE WIRE - **Properties:** High resistance per unit length, low melting point. - **Material:** Alloys of lead (Pb) and tin (Sn). - **Length:** Immaterial. - **Heating:** Temperature rises until heat produced equals heat lost by radiation. - **Current dependence:** $I \propto r^{3/2}$