Electrostatic Potential and Capacitance PDF May 2022 Past Paper

# Ch-2 Electrostatic Potential and Capacitance ## May 2022 ### Wednesday 11 **Introduction** - When an external force does work in taking a body from one point to another against a force like spring force or gravitational force, that work gets stored as the body's potential energy. - The sum of...

# Ch-2 Electrostatic Potential and Capacitance ## May 2022 ### Wednesday 11 **Introduction** - When an external force does work in taking a body from one point to another against a force like spring force or gravitational force, that work gets stored as the body's potential energy. - The sum of kinetic and potential energies is conserved. Forces of this kind are called conservative forces. - *eg.* Spring force and gravitational force. ## May 2022 ### Thursday 12 - Consider the field $E$ due to a charge $Q$ placed at the origin. Now imagine that we bring a test charge $q_o$ from a point R to a point P against the repulsive force on it due to the charge $Q$. - If $Q$ and $q_o$ are both positive or both negative, this will happen. **Note** - We assume the test charge $q_o$ so small that it doesn't disturb the original configuration, it means it has no electric field. ## May 2022 ### Friday 13 - In bringing the charge $q_o$ from R to P, we apply an external force $F_{ext}$ just enough to counter the repulsive electric force $F_E$. - $F_{ext} = -F_E$. - This means there is no net force on or acceleration of the charge $q_o$ when it is brought from R to P, i.e. it is brought with infinitesimally slow constant speed. In this situation, the work done by the external force is the negative of the work done by the electric force and gets fully stored in the form of potential energy of the charge $q_o$. ## May 2022 ### Saturday 14 - Electric potential difference is the amount of work done in moving a unit test charge from one point to the other in the against of electrostatic force without any acceleration. ## May 2022 ### Monday 16 - Force without any ac112. - Work ext. against a test charge. - $V_A - V_B = W_{external}$ - The test charge should be moved very slowly (No change in speed or K.E of $q_o$). - It is a scalar quantity and its SI unit is J/C. ## May 2022 ### Tuesday 17 - Electric Potential: - The amount of work done in bringing a unit test charge from infinity to that point in against of electrostatic force without any acceleration. - Electrostatic potential = workday charge. - $V = \frac{W}{q_o}$. - It is a scalar quantity and its SI unit is Volt. ## May 2022 ### Wednesday 18 - $1 J/C = 1 Volt$. - We assume $V_oo = 0$. - $V_A - V_{oo} = W_{oo \to A}$ - Potential at A $(V_A) = \frac{W_{oo \to A}}{q_o}$. - Some Important Points: - External agent shall be considered. This work is against electric force. ## May 2022 ### Thursday 19 - This work is path independent because electrostatic force is conservative. - $W_1 = W_2 = W_3$. - Work done by the conservative force in a round trip is zero. - $W_{A \to B \to A} = 0$. - Test charge would be very small so that it has no electric field. ## May 2022 ### Friday 20 - Potential due to a point charge. - Consider a point charge Q at the origin, take Q to be positive. The potential of a point P with position vector, distance r. $Q$ $2_o \to W_{ext}$ $2_0$ _____ _____ ____ r P r^2 $F = KQ2_o$ - Electric potential of point P. - $V_p - V_{oo} = W_{ext} cos \theta$ - $V_{oo} = 0 (let)$ $2_o$ ______ againsta electric force. ## May 2022 ### Saturday 21 - $V_p = \frac{W_{ext} cos \theta}{q_o}$. - Now we will calculate. - $W_{ext} P \to \infty$. - $W_{ext} \infty \to P = -W_{ext} P \to \infty$. - Force is conservative. - $F_{ext} = F_{elect} = K\frac{Q q_o}{r^2}$. - Work Done: - $\int_{P \to \infty}^{} F_E dr$ - $\int_{P \to \infty}^{} K\frac{Q q_o}{r^2} dr cos 180$ - $-KQ q_o \frac{-1}{r} |_{P}^{\infty}$ - $-KQ q_o (\frac{-1}{\infty} - \frac{-1}{P}$ - $-KQ q_o (\frac{-1}{P})$ ## May 2022 ### Monday 23 - $-KQ q_o (-\frac{1}{\infty} + \frac{1}{P})$ - $-KQ q_o (\frac{1}{P})$ - $- \frac{KQ q_o} {r}$ - $W_{ext} \infty \to P = -W_{ext} P \to \infty = - \frac{KQ q_o}{r}$ - $V_p = \frac{KQ q_o}{r} - \frac{KQ q_o}{r}$ - $V_p = \frac{KQ}{r} $ - $V_p = \frac{1}{4\pi \epsilon_o} \frac{Q}{r}$ - (take charge with sign) ## May 2022 ### Tuesday 24 - Potential of a point P due to multiple charges. - $V_p = \frac{KQ_1}{r_1} + \frac{KQ_2}{r_2} + \frac{KQ_3}{r_3} + ... + \frac{KQ_n}{r_n}$. - +-----+-----+-----+-----+ | | | | | | Q_1 | Q_2 | ... | Q_n | | | | | | +-----+-----+-----+-----+ - +-----+-----+-----+-----+ | | | | | | r_1 | r_2 | ... | r_n | | | | | | +-----+-----+-----+-----+ - $Q_1 + Q_2$ --------- a +-----+-----+-----+-----+ | | | | | | -Q_3 | | | | | | | | | +-----+-----+-----+-----+ - Where 18 V zero on the line joining two charges. - $V = \frac{KQ}{x} + \frac{K3Q}{a-x}$. - $0 = - \frac{1}{x} + \frac{3}{a-x}$. - $ \frac{3}{a-x}= \frac{1}{x}$. - $ 3x = a-x$ - $4x = a$. - $ x = \frac{a}{4}$. - And care. - $V_y = \frac{KQ}{y} - \frac{K3Q}{a+y}$. - $0 = \frac{1}{y} - \frac{3}{a+y}$. - $ \frac{3}{a+y} = \frac{1}{y}$. - $3y = a + y$. - $ 2y = a$. - $ y = \frac{a}{2}$. ## May 2022 ### Wednesday 25 - An electric dipole consists of two charges Q and -Q separated by a small distance 2a or 2l. It is total charge is zero. - Dipole moment (P) = 9 x 2a [Direction = -ve to +ve]. - Potential due to an electric dipole. - axis of dipole - it bisectors of dipole - of any point - potential on the axis of dipole. - Consider a positive point charge Q placed at the origin O, we wish to calculate its electro potential of a point P at a distance. ## May 2022 ### Friday 27 - Potential on the axis of dipole. - Consider an electric dipole consisting of two point charges -Q and +Q and separated by a distance 2a. Let P be a point on the axis of the dipole a distance r from its center. -Q +Q ___ ___ r - a r - Electric potential of point P due to the dipole: - $V = V_1 + V_2$. - $V = \frac{1}{4\pi \epsilon_o} \frac{-Q}{r-a} + \frac{1}{4\pi \epsilon_o} \frac{Q}{r+a}$. - $V = \frac{Q}{4\pi \epsilon_o} [\frac{1}{r+a} - \frac{1}{r-a}]$. - $V = \frac{Q}{4\pi \epsilon_o} [\frac{r-a - r - a}{(r+a)(r-a)}]$. - $V = \frac{Q}{4\pi \epsilon_o} [\frac{-2a}{r^2 - a^2}]$. - $V = - \frac{1}{4\pi \epsilon_o} \frac{2Qa}{r^2 - a^2}$. - $V = - \frac{1}{4\pi \epsilon_o} \frac{P}{r^2 - a^2}$ - P = Dipole Movement - P = 2Qa ## May 2022 ### Saturday 28 - Electric potential at point P due to the dipole. - $V = V_1 + V_2$. - $V = \frac{1}{4\pi \epsilon_o} \frac{-Q}{r-a} + \frac{1}{4\pi \epsilon_o} \frac{Q}{r+a}$. - $V = \frac{Q}{4\pi\epsilon_o} [\frac{1}{r+a} - \frac{1}{r-a}]$. - $V = \frac{Q}{4\pi\epsilon_o} [\frac{r-a - r - a}{(r+a)(r-a)}]$. - $V = \frac{Q}{4\pi \epsilon_o} [\frac{-2a}{r^2 - a^2}]$. - $V = - \frac{1}{4\pi\epsilon_o} \frac{2Qa}{r^2 - a^2}$. - $V = - \frac{1}{4\pi\epsilon_o} \frac{P}{r^2 - a^2}$. - $P = 2Qa$ - Now $a^2 << r^2$. - So $V = - \frac{1}{4\pi\epsilon_o} \frac{P}{r^2}$. ## May 2022 ### Monday 30 - Electric potential of an equatorial point of a dipole. - Consider an electric dipole consisting of charges -Q and +Q and separated by distance 2a. Let P be a point on the perpendicular bisector of the dipole at distance r from its center O. ## May 2022 ### Tuesday 31 - Electric potential of a point due to the dipole. - $V = V_1 + V_2$. - $V = \frac{1}{4\pi\epsilon_o} \frac{-Q}{\sqrt{r^2 + a^2}} + \frac{1}{4\pi\epsilon_o} \frac{Q}{\sqrt{r^2 + a^2}}$. - $V = - \frac{Q}{4\pi\epsilon_o\sqrt{r^2 + a^2}} + \frac{Q}{4\pi\epsilon_o\sqrt{r^2 + a^2}}$. - $V = 0$. ## June 2022 ### Wednesday 01 - Electric potential of any general point due to a dipole. - Consider an electric dipole consisting of two point charges -Q and +Q and separated by distance 2a. We wish to determine the potential of a point P at a distance r from the center O, the direction OP makes an angle θ with the dipole moment P. ## June 2022 ### Thursday 02 - Let AP = $r_1$ and BP = $r_2$ - Net potential of any point P due to the dipole is. - $V = V_1 + V_2$. - $V = \frac{1}{4\pi\epsilon_o} \frac{-Q}{r_1} + \frac{1}{4\pi\epsilon_o} \frac{Q}{r_2}$. - $V = \frac{Q}{4\pi\epsilon_o} [\frac{1}{r_2} - \frac{1}{r_1}]$. ## June 2022 ### Friday 03 - $\frac{Q}{4\pi\epsilon_o} [\frac{r_1-r_2}{r_1 r_2}]$. - If the point P lies faraway from the dipole: - $r_1 - r_2 = AB cos \theta$ - $r_1 - r_2 = 2a cos \theta$ - $r_1 + r_2 = r$. - $V = \frac{Q}{4\pi\epsilon_o} \frac{2a cos \theta}{r^2}$. - $V = - \frac{1}{4\pi\epsilon_o} \frac{P cos \theta}{r^2}$. - [P = 2Qa] ## June 2022 ### Saturday 04 - Relation between electric field and potential. - $V_B + \int_{B}^{A} Ed_r = V_A = V \implies E = -\frac{dV}{dr}$. - Work done to move the test charge from A to B is: - $W = F.dr$ - $W = -q_o E d_r$. - $W = charge X P.D$ - $W = q_o (V_B - V_A)$. - $W = q_o dV$. ## June 2022 ### Monday 06 - From (1) and (2), we get. - $-q_o E d_r = q_o dV$. - $E = - \frac{dV}{dr}$. - Now, $\frac{dV}{dr}$ is the rate of change of potential with distance, and is called potential gradient. - $d_V = - E.d_r$. - Integrating the above equation between points $r_1$ and $r_2$, we get: - when E field is uniform. ## June 2022 ### Tuesday 07 - $\int_{V_1}^{V_2}dV = - \int_{r_1}^{r_2} E . d_r$. - $V_1 - V_2 = - \int_{r_1}^{r_2} E . d_r$. - Along, - $E = E_x i + E_y j + E_z k$. - $dr = dx i + dy j + dz k$. - $dr = dx i + dy j + dz k$. - $dV = -E.dr$. - $dV = -(E_x i + E_y j + E_z k).(dx i + dy j + dz k)$. - $dV = -E_xdx - E_ydy - E_zdz$. - Partially diff. wirit x (y, z, const). - $\frac{dV}{dx} = - E_x$. - $dy = 0, dz = 0$. ## June 2022 ### Wednesday 08 - Similarly, - $\frac{dV}{dy} = - E_y, \frac{dV}{dz} = - E_z$. - Equipotential Surface: - Any surface that has same electric potential at every point on it is called an equipotential surface. - Example for a point charge. - Uncentric spheres with charge as centre of sphere are equipotential surfaces. ## June 2022 ### Thursday 09 - $V_1 = V_2 = V_3 = V_y = KQ/R$. +-----+-----+-----+-----+ | | | | | | V_1 | V_2 | V_3 | V_y | | | | | | +-----+-----+-----+-----+ +-----+-----+-----+-----+ | | | | | | R | R | R | R | | | | | | +-----+-----+-----+-----+ - Fixed capacitor. - Variable capacitor - Linear charge. - $V_1 = V_2 = V_3 = V_4$. +-----+-----+-----+-----+ | | | | | | V_1 | V_2 | V_3 | V_4 | | | | | | +-----+-----+-----+-----+ <start_of_image> +-----+-----+-----+-----+ | | | | | | ① | ② | ③ | ④ | | | | | | +-----+-----+-----+-----+ - Concentric cylinders with linear charge as axis of cylinders are equipotential surfaces. ## June 2022 ### Friday 10 - Properties of equipotential surface: - The potential difference between any two points on an equipotential surface is zero. - $V_A - V_B = 0$. - No work is done in moving a test charge over an equipotential surface. - $V_B - V_A = W_{external (agent b \to A)}$. - $W_1 = 0$. ## June 2022 ### Saturday 11 - The direction of $E$ is always perpendicular to equipotential surface. - $dV = -E. dr$. - $0 = -E.dr$. - $0 = E.dr cos \theta$. - $\theta = 90°$. - $E \perp dr$. - Equipotential surfaces are closer together in the regions of strong field and farther apart in the regions of weak field. ## June 2022 ### Monday 13 - No two equipotential surfaces can intersect each other. ## June 2022 ### Tuesday 14 - Electric Potential Energy: - The electric potential energy of a system of point charges may be defined as the amount of work done in assembling the charges at their locations by bringing them in from infinity. - Note: It is the energy possessed by a system of charges by virtue of their positions. When two like charges lie at infinite distance apart, their potential energy is zero because no work has to be done in moving one charge at infinite distance from the other. But when they are brought closer to one another, work has to be done against the force of repulsion. As electrostatic force is a conservative force, this extra work gets stored as the potential energy of the two charges. ## June 2022 ### Wednesday 15 - Potential energy of a system of two point charges. - Suppose a point charge $q_1$ is at rest at a given point $P_1$ in space. It takes no work to bring the first charge $q_1$ because there is no field yet to work against. - $W_1 = 0$. - Electric Potential due to charge $q_1$ at a point $P_2$ at a distance r from $P_1$ will be: - $V_1 = \frac{1}{4\pi\epsilon_o} \frac{q_1}{r}$. ## June 2022 ### Friday 17 - Now, if charge $q_2$ is moved in from infinity to point $P_1$, the work required is: - $W_2 = V_1 X q_2$. - $W_2 = \frac{1}{4\pi\epsilon_o} \frac{q_1 q_2}{r}$. - Potential energy of system. - $U = W_1 + W_2$. - $U = \frac{1}{4\pi\epsilon_o} \frac{q_1 q_2}{r}$. ## June 2022 ### Saturday 18 - Take, $q_1$ and $q_2$, with sign: - $q_1 q_2 > 0$, then +ve work has to be done. - $q_1 q_2 < 0$, then -ve work has to be done by external agent. - Potential energy of a system of multiple charges. - $U = U_{12} + U_{23} + U_{34} + .... + U_{n-1, n}$. ## June 2022 ### Monday 20 - Potential energy of a dipole in a uniform electric field. - Consider an electric dipole placed in a uniform electric field E with its dipole moment P making an angle θ with the field. - Two equal and opposite forces +QE and -QE act on its two ends. - Two forces form a couple. - The torque exerted by the couple will be: - $\tau = PE sin \theta$. - If dipole is rotated through a small and dθ against the torque acting on it, then the small work done is: - $dW = τd \theta$. - $dW = PE sin \theta d \theta$. - Now Total work done in rotating the dipole from an angle $θ_1$ to $θ_2$ - $\int_{θ_1}^{θ_2} dW = \int_{θ_1}^{θ_2}PE sin \theta d \theta$. - $W = PE [-cos \theta]_{θ_1}^{θ_2}$. - $W = PE [cos θ_1 - cos θ_2]$. - $U = PE [cos θ_1 - cos θ_2]$. ## June 2022 ### Friday 24 - Capacitor. - Some electric devices require very high current to start them (Static friction) like fan, motor etc. - Capacitor = a conductor which stores electrical charge or electrical energy and supply it at once. - Capacitance = The electrical capacitance of a conductor is the measure of its ability to hold electric charge. - $Q \propto V$. - $Q = CV$. - $C = \frac{Q}{V}$. - C is a capacitance of a conductor. - C is independent of Q and V. - If depends upon the following factors: - Size and shape of the conductor. - Nature (permittivity) of the surrounding medium. - Presence of the other conductor in its neighborhood. - Unit of C = $\frac{Q}{V}$ = Farad. - 1 F = 1 Farad. - Dimensional Formula: - $C = \frac{Q}{V} = \frac{IT}{ML^2T^{-3}} = [M^{-1}L^{-2}T^4A^2]$. ## June 2022 ### Saturday 25 - Pictorial representation of a capacitor. - Fixed capacitor. - Variable capacitor. - Parallel plate capacitor: - Two large plane parallel conducting plates, separated by a small distance. ## July 2022 ### Friday 01 <start_of_image>- **G = 260** +-----+-----+ | | | | + | - | | | | +-----+-----+ | | | | + | - | | | | +-----+-----+ | | | | + | - | | | | +-----+-----+ | | | | + | - | | | | +-----+-----+ K d | | | | | | | | | | +-----+-----+ | | | | - | + | | | | +-----+-----+ | | | | - | + | | | | +-----+-----+ | | | | - | + | | | | +-----+-----+ | | | | - | + | | | | +-----+-----+ $d << Area$ - Infinite Plate. - Let, A= Area of each plate. - d = distance between the two plates. - σ = uniform surface charge density on the two plates. - Q = ± σA = total charge on each plate. ## July 2022 ### Saturday 02 - The net electric field inside the plates. - $E_{net} = E_1 + E_2$. - $E_{net} = 2\sigma/ \epsilon_o + 2 \sigma / \epsilon_o$. - $E_{net} = 4\sigma / \epsilon_o$. - $E_{net} = \frac{Q}{A \epsilon_o}$. - P.D between the plates: - $ΔV = E_{ext} d$. - $ΔV = \frac{Q}{A \epsilon_o} d$. - $ΔV = \frac{\sigma d}{\epsilon_o}$. - $C = \frac{Q}{ΔV} = \frac{Q}{\sigma d/ \epsilon_o} = \frac{\epsilon_o A}{d}$. ## July 2022 ### Monday 04 - $C = \frac{\epsilon_o A}{d}$. - Now we can conclude that capacitance depends on: - Area of the plates (C α A). - Distance b/w the plates (C α 1/d). - Permittivity of the medium between the plates (C α ε). - Energy stored in a capacitor: - +-----+ | | | +Q | | | +-----+ $d_q$ +-----+ | | | -Q | | | +-----+ - +-----+ | | | +Q | | | +-----+ +-----+ | | | -Q | | | +-----+ - +-----+ | | | +Q | | | +-----+ +-----+ | | | -Q | | | +-----+ - $U = \frac{1}{2} CV^2 = \frac{1}{2} \frac{Q^2}{C} = \frac{1}{2} Q ΔV$. ## July 2022 ### Tuesday 05 - Consider a capacitor of capacitance C. Suppose the charge is transferred from plate 2 to plate 1 bit by bit, so that at the end, conductor 1 gets +Q charge and conductor 2 gets -Q charge. Suppose now a small charge dQ be transferred from plate 2 to 1. Work done: - $dW = V dQ$ - $dW = \frac{Q}{C} dQ$ - $W_{ext(agent)} = \int_{}^{} dW =\int_{0}^{Q} \frac{Q}{C} dQ$. - $W_{ext(agent)} = \frac{1}{C} \int_{0}^{Q} QdQ$. - $W{ext(agent)} = \frac{1}{C} [\frac{Q^2}{2}]_{0}^{Q}$. - $W_{ext(agent)} = \frac{Q^2}{2C}$. - - $U = \frac{Q^2}{2C}$. ## July 2022 ### Wednesday 06 - $U = \frac{Q^2}{2C}$. - $U = \frac{1}{2} CV^2$. - In terms of Q and V: - $C = \frac{Q}{V}$ - $U = \frac{1}{2} Q V$. - In terms of V and C: - $Q = CV$. - $U = \frac{1}{2} CV^2$. - $U = \frac{1}{2} \frac{Q^2}{C} = \frac{(\epsilon_o A)^2d}{2C^2} = \frac{1}{2} \frac{(\epsilon_o A)^2d\sigma A}{\epsilon_o A} = \frac{1}{2} \sigma d\sigma A$ - $U = \frac{1}{2} \epsilon_o E^2 d A$. ## July 2022 ### Thursday 07 - $E = \frac{\sigma}{\epsilon_o}$. - $U = \frac{1}{2} \epsilon_o E^2 A.d$. - Combination of capacitors in series and in parallel. - Capacitance in series: - When the negative plate of one capacitor is connected to the positive plate of the second and the negative of the second to the positive of the third, and so on, the capacitors are said to be connected in series. - $C_1 - C_2 - C_3$. ## July 2022 ### Friday 08 - To be connected in series. - $C_1 - C_2 - C_3$ - $+Q - Q + Q - Q + Q - Q$ - +-----+-----+-----+ | | | | | -V1 | -V2 | -V3 | | | | | +-----+-----+-----+ - Consider three capacitors of capacitance $C_1, C_2$ and $C_3$ connected in series. A P.D $V$ is applied across the combination. This set up charges ±Q on the two plates of each capacitor. - P.D across the capacitors. - $V_1 = \frac{Q}{C_1}$, $V_2 = \frac{Q}{C_2}$, $V_3 = \frac{Q}{C_3}$. ## July 2022 ### Saturday 09 - Now let the net PD be V. - $V = V_1 + V_2 + V_3$. - $V = \frac{Q}{C_1} + \frac{Q}{C_2} + \frac{Q}{C_3}$. - $V = \frac{Q (C_1+C_2+C_3)}{C_1 C_2 C_3}$. - $\frac{Q}{V} = \frac{C_1 C_2 C_3}{C_1 + c_2 + C_3}$. - $\frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3}$. - C is the equivalent capacitance of the series combination. ## July 2022 ### Monday 11 - Capacitors in parallel. - When the positive plates of all capacitors are connected to one common point, and the negative plates to another common point, the capacitors are said to be connected in parallel. +-----+-----+-----+ | | | | | +Q_1 | +Q_2 | +Q_3 | | | | | +-----+-----+-----+ $C_1$ +-----+-----+-----+ | | | | | -Q_1 | -Q_2 | -Q_3 | | | | | +-----+-----+-----+ $C_2$ +-----+-----+-----+ | | | | | +Q_1 | +Q_2 | +Q_3 | | | | | +-----+-----+-----+ $C_3$ +-----+-----+-----+-----+ | | | | | | -Q_1 | -Q_2 | -Q_3 | | | | | | | +-----+-----+-----+-----+ ## July 2022 ### Tuesday 12 - Consider three capacitors of capacitance C1, C2, and C3 connected in parallel. A potential difference V is applied across the combination. - All the capacitors have a common P.D V but different charges. - $Q_1 = C_1V, Q_2 = C_2V, Q_3 = C_3V$. - Total charge stored in the combination: - $Q = Q_1 + Q_2 + Q_3$. ## July 2022 ### Wednesday 13 - $Q = C_1 V + C_2 V + C_3V$. - $C_p V = (C_1+C_2+C_3)V$. - Where $C_p$ is the equivalent capacitance of the parallel combination. - $C_p = \frac{C_1 C_2 C_3}{C_1+C_2+C_3}$. - $C_p = C_1 + C_2 + C_3$. - Effect of dielectric on capacitance: ## July 2022 ### Thursday 14 - Dielectrics: - Dielectrics are non-conducting substances which transmit electric effect without actually conducting electricity. - Dielectrics are of two types: - Non Polar. - Polar. - Non polar dielectrics: - The molecules in which the center of positive charge coincides with the centre of negative charge are called non-polar molecules. Such molecules are symmetric in shape. - *eg.* $O_2$, $CO_2$. - +-----+ | | | H | | | +-----+ +-----+ | | | | | |