Electric Potential and Capacitance Formulas- Physics Formula
1. Electric Potential Difference
Scalar quantity
`V_A-V_B=\frac{W_{ext_{B\rightarrow A}}}{q_o}`
2. Electric Potential at a Point
`V_A=\frac{W_{ext_{\infty\rightarrow A}}}{q_0}`
3. Potential due to a Point charge
`V=\frac Q{4\pi\varepsilon_0R}=\frac{KQ}R`
4. Potential due to a uniformly charges ring
At axial Point
`V_{AP}=\frac{KQ}{\sqrt{R^2+x^2}}`
At centre
`V_{centre}=\frac{KQ}R`
5. Potential due to Electric dipole
At Axial point
`V_{ax}=\frac{KP}{x^2}`
At Perpendicular bisector Or Equatorial point
`V_{eq}=0`
At General Point
`V_x=\frac{KP\cos\theta}{x^2}`
6. Relation between Fields and Potential
"Potential always decreases in direction of field"
`\|\triangle V\|=E.\triangle r`
7. `E=-\frac{\partial V}{\partial x}\hat i+\left(-\frac{\partial V}{\partial y}\right)\hat j+\left(-\frac{\partial V}{\partial z}\right)\hat k`
8. Potential due to a Hollow [Conducting And Non-Conducting] and Solid Conducting Sphere.
`V_{inside}=V_{surf}=\frac{KQ}R`
`V_{out}=\frac{KQ}r`
Where R = Radius of Sphere
And r= distance from centre of sphere
9. Potential due to solid Non-Conducting sphere[Dielectric/Insulated]
Inside (x<R)
`V_i=\frac{KQ\(3R^2-x^2\)}{2R^3}`
Centre
`V_c=\frac{3KQ}{2R}`
Surface
`V_s=\frac{KQ}R`
Outside sphere (x>R)
`V_o=\frac{KQ}r`
10. Earthing
Connect any conductor to earth
`V_{cond}=0`
Charge need not to be zero.
11. Electrostatic energy (U)
`U=\frac{KQ_1Q_2}r`
Number of Pairs =n(n-1) /2
`U_f-U_i=W_{ext}=-W_{cons}`
12. Potential energy of an Electric Dipole in uniform Electric Field.
`U_\theta=-PE\cos\theta`
`W_{ext}=U_f-U_i=PE(cos\theta_1-\cos\theta_2)`
`theta`=90° , U=0 , `tau`=PE (maximum)
`theta`=0° , U=-PE (minimum), `tau`=0 (stable equilibrium)
`theta`=180° , U=+PE (maximum), `tau`=0 (unstable equilibrium)
13. Capacitor
`C=\frac qV`
Unit = Farad
C is independent of q and V.
It depends on dimension of conductor and property of medium.
14. Spherical Conductor
`C=4\pi\varepsilon_0R`
15. Energy stored
`U=\frac{CV^2}2=\frac{q^2}{2C}`
16. Cylindrical Capacitor
`C=\frac{2\pi\varepsilon_0l}{\ln\{\frac{R_2}{R_1}}\}`
17. Parallel Plate Capacitor
`C=\frac{\varepsilon_0A}d`
Force on one plate due to other plate.
`F=\frac{Q^2}{2A\varepsilon_0}`
18. Combination of Capacitor
Series combination
`\frac1{C_{eq}}=\frac1{C_1}+\frac1{C_2}+..`
Parallel Combination
`C_{eq}=C_1+C_2+..`
19. Dielectric in a Capacitor (K)
`C'=KC_0`
`E'=\frac{E_0}K`
`V'=\frac{V_0}K`
`C=\frac{\varepsilon_0A}{{\frac{t_1}{K_1}}+{\frac{d-t}{K_2}}}`
20. Circuit with Capacitor and Resistor
At t= 0,
Replace Capacitor ➡ straight wire
At t= infinite (steady state)
Replace Capacitor with broken wire.
21. Energy Density
`E_d=\frac U{Volume}=\frac U{Ad}=\frac{\varepsilon_0E^2}2`
22. Charging of Capacitor
Growth of current
`Q=Q_0(1-e^{-\frac t\tau})`
`i=i_0e^{-\frac t\tau}`
Where `tau`= time constant = RC
23. Discharging of Capacitor
Decay of current
`q=q_0e^{-\frac t\tau}`
`i=i_0e^{-\frac t\tau}`