Here students can locate TS Inter 2nd Year Physics Notes 5th Lesson Electrostatic Potential and Capacitance to prepare for their exam.
TS Inter 2nd Year Physics Notes 5th Lesson Electrostatic Potential and Capacitance
→ Coulomb force between two stationary charges is also a conservative force, i.e., work done in moving a test charge depends only on initial and final positions but not on the path.
→ Potential due to a point charge q’ at a distance ‘r’ V(r) = \(\frac{1}{4 \pi \varepsilon_0} \cdot \frac{Q}{r}\)
→ Potential due to a system of charges :
Consider a system of charges q1, q2, q3, ………… qn then potential at a point P = the algebraic sum of individual potentials at that point.
V = V1 + V2 + V3 + ……….. + Vn
or
V = \(\frac{1}{4 \pi \varepsilon_0}\left(\frac{\mathrm{q}_1}{\mathrm{r}_1}+\frac{\mathrm{q}_2}{\mathrm{r}_2}+\frac{\mathrm{q}_3}{\mathrm{r}_3}+\ldots \ldots+\frac{\mathrm{q}_{\mathrm{n}}}{\mathrm{r}_{\mathrm{n}}}\right)\)
→ Potential on a conducting sphere :
When a charge q’ is given to a conducting sphere it will be uniformly distributed over the sphere of radius R’.
(a) Potential at a distance ‘r’
(V) = \(\frac{1}{4 \pi \varepsilon_0} \frac{\mathrm{q}}{\mathrm{r}}\). (r > R)
(b) For a point inside the shell the potential is constant V = \(\frac{1}{4 \pi \varepsilon_0} \frac{q}{R}\)
→ Equipotential surface :
An equipotential surface has a constant potential at all points on that surface.
Ex: Potential at a distance V from charge q’
(V) = \(\frac{1}{4 \pi \varepsilon_0} \frac{\mathrm{q}}{\mathrm{r}}\). Now a circle of radius r from q’ will have same potential i.e., that
circle represents an equipotential circle.
Note Work done in moving a charge q’ on an equipotential surface is zero.
→ Potential energy due to a system of charges:
Consider two charges say q1 and q2 placed at positions r1 and r2 from origin. Then distance between them = r12.
(a) Potential energy of the system of two charges
U = \(\frac{1}{4 \pi \varepsilon_0} \frac{q_1 q_2}{r_{12}}\)
(b) If a system has three charges say q1 q2 and q3 at positions r1, r2 and r3 then potential energy of that system = the alzebraic sum of potential energy between every two charges.
∴ Potential energy of system U = \(\frac{1}{4 \pi \varepsilon_0}\left[\frac{q_1 q_2}{r_{12}}+\frac{q_2 q_3}{r_{23}}+\frac{q_3 q_1}{r_{13}}\right]\)
→ Potential energy of a dipole in an electric field :
When an electric dipole is placed in a uniform electric field. Net force on the charges are Eq and – Eq. So resultant force is zero. But dipole will experience some torque τ on it. So it will rotate say from position θ1 to θ2 with the field (E). Torque τ = workdone by external field E
W = PE (cos θ0 – cos θ1) where
p = dipolemoment.
→ Electrostatic potential is constant through out the volume of the conductor and has same value equal to that on the surface.
→ Electric field at the surface of a charged a conductor E = \(\frac{\sigma}{\varepsilon_0}\)n .
→ Dielectrics : Dielectrics are the substances which does not conduct electricity.
→ Dielectric constant is defined as the ratio of force between two charges in vacuum to force between the same charges with a dielectric between them when distance is constant.
K = \(\frac{\mathrm{F}_1}{\mathrm{~F}}=\frac{1}{4 \pi \varepsilon_0} \frac{\mathrm{q}_1 \mathrm{q}_2}{\mathrm{r}^2} / \frac{1}{4 \pi \varepsilon} \frac{\mathrm{q}_1 \mathrm{q}_2}{\mathrm{r}^2}\)
K = \(\frac{\varepsilon}{\varepsilon_0}\) = εr
where k = relative permittivity (or) dielectric constant.
→ Polar dielectrics: A dielectric molecule in which + ve and – ve charges are polarised into two groups without application of electric field is called polar dielectric field.
→ Non-polar dielectrics : A dielectric molecule in which the centres of all positive charges and all negative charges coincide when there is no electric field is called non polar dielectric.
→ In external electric field E’ behaviour of polar and non-polar dielectrics is same.
→ When a dielectric of constant k’ is introduced in an electric field ‘E’ then intensity of electric field inside the dielectric is reduced to E/K i.e., Ed = E/K.
→ Electric susceptibility (χ) : It is defined as the ratio of dipolemoment per unit volume to Electric field E
χ = \(\frac{P}{E}\) where P is polarisation i.e., dipole moment per unit volume.
→ Capacity (Q : It is defined as the ratio of charge Q’ to potential (V) of a conductor.
Capacity (C) = \(\frac{\text { Charge }(Q)}{\text { Potential }(V)}\) unit: Farad
Note: Capacity of a conductor depends on its geometrical shape and medium between the plates.
→ Parallel plate capacitor: It consists of two metallic plates separated by some distance ’d’ and one plate is connected to earth.
Capacity of parallel plate capacitor s0A
C = \(\frac{\varepsilon_0 \mathrm{~A}}{\mathrm{~d}}\) (with vacuum between the plates)
C = K \(\frac{\varepsilon_0 \mathrm{~A}}{\mathrm{~d}}\) (with a dielectric K filled between the plates).
→ Energy stored in a capacitor (U) : When a capacitor is charged to a potential ‘V’ it will store some electrical charge in it. While dis-charging the capacitor this electric charge is capable of doing some work. So while charging the work done to charge the capa-citor is stored in the form of potential energy in it.
Energy stored in a capacitor
U = \(\frac{1}{2}\)CV2 = \(\frac{\mathrm{QV}}{2}=\frac{\mathrm{Q}^2}{2 \mathrm{C}}\)
→ Energy density (OR) Energy stored per unit volume : It is defined as the ratio of energy stored in a capacitor to the total volume between the plates of a capacitor.
Energy density U = \(\frac{1}{2}\)ε0E2
Where E is intensity of electric field between the plates.
→ Van deGraaff generator: It is used to produce very high voltages such as 1 million volts.
→ Force between two charges in free space is
F = \(\frac{1}{4 \pi \varepsilon_0} \frac{\mathrm{q}_1 \mathrm{q}_2}{\mathrm{r}^2}\)
→ Relative permittivity εr = \(\frac{\varepsilon}{\varepsilon_0}\)
\(\frac{1}{4 \pi \varepsilon_0}\) = 9 × 109 N – m2/C2
ε0 = 8.854 × 10-12 F/m or \(\frac{c^2}{N-m^2}\)
→ From the superposition principle, the resultant force on a given charge due to multiple charges is \(\overline{\mathrm{F}}_{\mathrm{R}}=\overline{\mathrm{F}}_1+\overline{\mathrm{F}}_2 \ldots \ldots \ldots \overline{\mathrm{F}}_{\mathrm{n}}\) (Vector sum of forces)
→ Intensity of electric field E = \(\frac{\mathrm{F}}{\mathrm{q}}=\frac{1}{4 \pi \varepsilon_0} \frac{\mathrm{q}}{\mathrm{r}^2}\)
→ Force experienced by a charge q in electric field E is F = Eq.
→ Potential due to a point charge is V = \(\frac{1}{4 \pi \varepsilon_0} \frac{\mathrm{q}}{\mathrm{r}}\)
→ Relation between Intensity of electric field E̅ and potential V is E = \(\frac{V}{d}\).
→ Workdone to move a charge q through a potential V is W = qV.
→ Electrostatic potential due to a system of charges is
VR = V1 + V2 + V3 + ……….. + Vn
(algebraic sum of individual potentials.)
→ Dipoles: A dipole consists of two equal and opposite charges separated by a distance 2a.
- Dipolemoment (p) = q2a = 2qa.
- Potential due to a dipole at any point
V = \(\frac{1}{4 \pi \varepsilon_0} \frac{2 \mathrm{qa} \cos \theta}{\mathrm{r}^2}=\frac{\mathrm{p} \cos \theta}{4 \pi \varepsilon_0 \mathrm{r}^2}=\frac{\overline{\mathrm{p}} \cdot \overline{\mathrm{r}}}{4 \pi \varepsilon_0 \mathrm{r}^2}\)
where r̅ is a unit vector along the line joining centre of dipole and given point. - Torque on dipole in uniform electric field E is τ = p̅. E̅ = |p̅||E̅| sin θ
- Work done to rotate a dipole from angle θ1 to θ2 in electric field W = potential energy stored U(θ) = PE (cos θ1 – cos θ2)
→ Total electric flux through given area
\(\oint_E=\oint_s \bar{E} \cdot \overline{d s}=\oint_s\) E ds cos θ
From Gauss’s theorem \(\oint_E=\oint_s \bar{E} \cdot \overline{d s}\)E̅.ds̅ = \(\frac{\mathrm{q}}{\varepsilon_0}\)
Linear charge density λ = \(\frac{\mathrm{dq}}{\mathrm{dl}}\) where dq is charge on the infinitesimal length dl.
→ For an infinitely long straight charged conductor:
(a) Intensity at any perpendicular point is E = \(\frac{\lambda}{2 \pi \varepsilon_0 r}\)
(b) Potential at any perpendicular point is V = \(\frac{\lambda}{2 \pi \varepsilon_0}\)loge r + K
→ For a charged infinite plane sheet:
Intensity of electric field at a distance r from the plane is E = \(\frac{\sigma}{2 \varepsilon_0}\)
→ For a charged spherical shell : Surface charge density σ = \(\frac{\mathrm{dQ}}{\mathrm{dS}}\) or
σ = \(\frac{\text { Charge } \mathrm{q}}{\text { Surface area } \mathrm{S}}\)
(i) For points outside the shell
(a) Intensity of electric field E = \(\frac{\sigma}{s_0} \frac{\mathrm{R}^2}{\mathrm{r}^2}\)
(b) Potential V = \(\frac{1}{4 \pi \varepsilon_0} \frac{q}{r}\) + K where K is a constant of Integration.
(ii) For points on the surface of the sphere
(a) Intensity of electnc field E = \(\frac{1}{4 \pi \varepsilon_0} \frac{\mathrm{q}}{\mathrm{R}^2}\)
(b) Potential V = \(=\frac{1}{4 \pi \varepsilon_0} \frac{q}{R}\)
(iii) For points inside the shell
(a) Intensity of electric field E = O;
(b) Potential V = \(=\frac{1}{4 \pi \varepsilon_0} \frac{q}{R}\)
→ Capacity C = \(\frac{\text { Charge q }}{\text { Potential V }}\)
C = \(\frac{\mathrm{K} \varepsilon_0 \mathrm{~A}}{\mathrm{~d}}\) =KC0 with dielectric of constant K.
→ By placing a dielectric between the plates of a capacitor
- Intensity of electric field EM = \(\frac{\mathrm{E}_0}{\mathrm{~K}}\)
- Capacity C = KC0
- Potential VM = \(\frac{\mathrm{V}_0}{\mathrm{~K}}\)
→ When a dielectric is partially introduced between the plates of a capacitor then its capacity
C = \(\frac{\varepsilon_0 A}{d-t+\frac{t}{K}}\) or C = \(\frac{\varepsilon_0 \mathrm{~A}}{\mathrm{~d}-t\left(1-\frac{1}{\mathrm{~K}}\right)}\)
decrease in distance = d – t\(\left(1-\frac{1}{\mathrm{~K}}\right)\) and ‘t’ is the thickness of the dielectric slab.
→ In parallel combination of capacitors, the resultant capacity C = C1 + C2 + C3 + ……… etc.
→ When capacitors are connected in series, the resultant capacity is given by
\(\frac{1}{C}=\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}\)….or C = \(\frac{\mathrm{C}_1 \mathrm{C}_2}{\mathrm{C}_1+\mathrm{C}_2}\)
→ Energy stored in a capacitor
U = \(\frac{1}{2}\)CV2 = \(\frac{\mathrm{Q}^2}{2 \mathrm{C}}=\frac{\mathrm{QV}}{2}\)
→ Effect of dielectric on a capacitor :
1. When charging battery is continued
(a) Potential V = V0;
(b) Capacity C = KC0
(c) Charge Q = KQ0;
(d) Energy stored U = KU0
where V0, C0, Q0 and U0 are values without dielectric.
2. When charging battery is removed from circuit
(a) Potential V = \(\frac{\mathrm{V}_0}{\mathrm{~K}}\);
(b) Capacity C = KC0
(c) Charge on capacitor q = q0
(d) Energy stored U = \(\frac{\mathrm{U}_0}{\mathrm{~K}}\)