- Electric potential is a scalar quantity, and it is positive near a positive charge and negative near a negative charge.
- Electric potential is taken as zero at infinity because the force between charges becomes zero at infinite separation.
- The potential difference between two points is measured using a voltmeter, which is connected in parallel with the circuit, with its positive terminal at the higher-potential point.
Definitions [24]
Definition: Electric Potential
Electric potential at a point is the work done in bringing a unit positive test charge from infinity to that point in an electric field.
The SI unit of electric potential is the volt.
Definition: Potential Difference
The difference in electrical potentials between two points is known as potential difference or voltage.
The unit of potential difference or potential is joule/coulomb, called a volt (V).
Define Electric potential.
Electric potential is a measure of work done on the unit's positive charge to bring it to that point against all electrical forces. It is represented as ‘V’.
Definition: Potential at a Point
The potential at a point is defined as the amount of work done per unit charge in bringing a positive test charge from infinity to that point.
Definition: Potential Difference
The potential difference (p.d.) between two points is equal to the work done per unit charge in moving a positive test charge from one point to the other.
OR
The work done per unit positive charge in moving a charge from one point to another in an electric field is called the potential difference between those two points.
Define the following:
Potential difference
Potential difference: The potential difference between two points may be defined as the work done in moving a unit positive charge from one point to the other.
Definition: Electric Potential Due to a Point Charge
The work done by an external agent in bringing a unit positive test charge slowly from infinity to a point in an electric field, against the electrostatic force, is called the electric potential at that point.
Definition: Equipotential Surface
The surface at which electric potential is the same at each point is called an equipotential surface.
Definition: Electric Dipole
An electric dipole is a pair of equal and opposite charges separated by a small distance.
Definition: Dipole Length
If the charges are separated by a distance 2a2a, then 2a2a is called the dipole length.
Definition: Electrostatic Equilibrium
The condition in which charges in a conductor are at rest, and no further motion of charges occurs.
Definition: Electrostatic Shielding
The phenomenon in which the electric field inside a cavity of a conductor is zero, irrespective of external charges or fields, is called electrostatic shielding.
Definition: Surface Charge Density
Surface charge density is the charge per unit area on the surface of a conductor and is denoted by \[\sigma\].
Definition: Equipotential Body
A conductor in electrostatic equilibrium is an equipotential body, meaning all points on it are at the same electric potential.
Definition: Dielectrics
Non-conducting substances which cannot transmit electric charge through them are called dielectrics.
Definition: Polar Molecule
The molecule in which the centres of positive and negative charges are separated even when there is no external field, and which has a permanent dipole moment, is called a polar molecule. (e.g. HCl, H₂O, alcohol, NH₃)
Definition: Non-polar Molecule
The molecule in which the centres of positive and negative charges coincide and which has no permanent dipole moment in its normal state is called a non-polar molecule. (e.g. O₂, H₂, N₂, CO₂, benzene, methane)
Definition: Polar Dielectric
A dielectric that has a permanent electric dipole moment even if the external electric field is absent is called a polar dielectric.
Definition: Non-polar Dielectric
A dielectric in which every molecule has zero dipole moment in its normal state is called a non-polar dielectric.
Definition: Electric Polarisation
Alignment of dipole moments (permanent or induced) in the direction of an applied electric field is called polarisation.
Definition: The Parallel Plate Capacitor
A capacitor that consists of two large, parallel, conducting plates separated by a small distance is called a parallel plate capacitor.
Definition: Equivalent Capacitance
The capacitance of a single capacitor that stores the same charge at the same voltage as the entire combination is called the equivalent capacitance of the combination.
Definition: Potential Difference (V)
The work done per unit charge in moving a charge from one plate of a capacitor to the other is called the potential difference between the plates.
Definition: Energy Stored in a Capacitor
The work done in the transfer of charge q between the two plates of a capacitor, which gets stored in the form of potential energy of the system, is called the energy stored in a capacitor.
Formulae [17]
Formula: Potential Due to a Dipole
\[V=\frac{q(2a\cos\theta)}{4\pi\varepsilon_0(r^2-a^2\cos^2\theta)}\]
if r ≫ a:
\[V=\frac{p\cos\theta}{4\pi\varepsilon_0r^2}\]
Formula: Potential at a Point Due to a System of Charges
\[V=\frac{1}{4\pi\varepsilon_0}\sum_{i=1}^n\frac{q_i}{r_i}\]
Formula: Electric Potential at a Point
V = \[\frac {W}{Q}\]
or
W = QV
Formula: In a medium of dielectric constant K K
\[V(r)=\frac{1}{4\pi\varepsilon_0K}\frac{q}{r}\]
- V(r) = electric potential at distance rr from the charge
- q = source charge
- ε0 = permittivity of free space
- K = dielectric constant of medium
- Reference is taken such that V(∞) = 0.
Formula: Potential Due to a Point Charge
\[V=\frac{Q}{4\pi\varepsilon_0r}\]
Potential due to System of Charges:
\[U=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r_{12}}\]
Formula: Electric Potential due to a Point Charge
V = \[\frac{1}{4\pi\varepsilon_0}\cdot\frac{q}{r}\]
Varies on spherical shell carrying charge q and radius R:
- Inside shell (r < R): V = \[\frac {1}{4πε_0}\] ⋅ \[\frac {q}{R}\]
- On surface (r = R): V = \[\frac {1}{4πε_0}\] ⋅ \[\frac {q}{R}\]
- Outside shell (r > R): V = \[\frac {1}{4πε_0}\] ⋅ \[\frac {q}{r}\]
Formula: Electric Potential Energy of Two Point Charges
U = \[\frac{1}{4\pi\varepsilon_0}\cdot\frac{q_1q_2}{r_{12}}\]
Formula: Work Done on an Equipotential Surface
When a charge q0 is moved from point A to point B on the same equipotential surface:
W = q0(VA − VB)
Since VA = VB on the surface:
W = 0
Formula: Potential Due to an Electric Dipole
\[V=\frac{1}{4\pi\varepsilon_{0}}\cdot\frac{p\cos\theta}{r^{2}}=\frac{1}{4\pi\varepsilon_{0}}\cdot\frac{\vec{p}\cdot\vec{r}}{r^{3}}(r>>a)\]
Formula: Electric Dipole Moment
The electric dipole moment is:
\[\vec{p}=q(2a)\hat{p}\]
Its direction is from the negative charge to the positive charge.
Formula: Electric Field on a Charged Conductor Surface
\[\vec{E}=\frac{\sigma}{\varepsilon_0}\hat{n}\]
where
- σ = surface charge density
- \[\hat n\] = outward normal unit vector
- \[\varepsilon_0\] = permittivity of free space.
Magnitude form:
E = \[\frac{\sigma}{\varepsilon_0}\]
Vector form:
$$\vec{E} = \frac{\sigma}{\varepsilon_0}\hat{n}$$
Formula: Polarisation Vector (P)
Defined as dipole moment per unit volume:
\[P=\frac{\text{dipole moment}}{\mathrm{volume}}=np\]
Formula: Capacitance of a Parallel Plate Capacitor
For two plates separated by distance d:
\[C=\frac{\varepsilon_0A}{d}\]
With a dielectric medium:
\[C=\frac{K\varepsilon_0A}{d}\]
Formula: Series Combination
\[{\frac{1}{C_S}=\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}+\cdots}\]
For n identical capacitors of capacitance C each: CS = \[\frac {C}{n}\]
Formula: Voltage Distribution (Special Formula)
For two capacitors in series, the voltage across each is:
\[V_1=\frac{C_2}{C_1+C_2}\cdot V\]
\[V_2=\frac{C_1}{C_1+C_2}\cdot V\]
Physical Insight: The smaller the capacitor, the larger the voltage drop across it in a series combination. This is why identical series capacitors share voltage equally.
Formula: Parallel Combination
\[{C_P=C_1+C_2+C_3+\cdots}\]
For n identical capacitors of capacitance C each: CP = nC
Physical Insight: Adding capacitors in parallel is like adding more storage tanks — the total storage capacity simply increases.
Formula: Energy Stored / Work Done in a Capacitor
W = \[\frac {1}{2}\]qV
OR
U = \[\frac {Q^2}{2C}\] = \[\frac {1}{2}\]QV = \[\frac {1}{2}\]CV2
SI unit: Joule (J)
Key Points
Key points: Potential and Potential Difference
Key Points: Electric Potential Due to a Point Charge
- Electric potential at a point is the work done per unit positive test charge in bringing it slowly from infinity to that point, against the electric field.
- For a point charge q in air/vacuum:
V(r) = \[\frac{1}{4\pi\varepsilon_0}\frac{q}{r}\] - In a medium of dielectric constant K:
V(r) = \[\frac{1}{4\pi\varepsilon_0K}\frac{q}{r}\] - Positive charge produces positive potential; negative charge produces negative potential.
- Potential due to a point charge is spherically symmetric and depends only on distance r.
- Distance dependence:
F ∝ 1/r2, E ∝ 1/r2, V ∝ 1/r. - The potential at infinity is taken as zero; only potential differences are physically significant.
- The electrostatic field is conservative, so the work done in moving a charge between two points is path independent.
Key Points: Combination of Capacitors
Capacitors in Series:
Equivalent capacitance: \[\frac{1}{C_s}=\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}+\cdots\]
- Same voltage (V) across all capacitors
- Charge divides
- The equivalent capacitance is greater than the largest capacitor
Capacitors in Parallel:
\[C_p=C_1+C_2+C_3+\cdots\]
- Same voltage (V) across all capacitors
- Charge divides
- The equivalent capacitance is greater than the largest capacitor
Important Questions [8]
- Obtain an Expression for Electric Potential ‘V’ at a Point in an End-on Position I.E. Axial Position of the Electric Dipole.
- Calculate electric potential at a point P which is at a distance of 9 cm from a point charge of 50 μC.
- Define Equipotential Surface.
- What is meant by an equipotential surface?
- Three Capacitors of Capacitance C 1 = 3 μ F , C 2 = 6 μ F , C 3 = 10 μ F , Are Connected to a 10v Battery as Shown in Figure 3 Below :
- Figure 4 Below Shows a Capacitor C, an Inductor L and a Resistor R, Connected in Series to an A.C. Supply of 220 V
- Deduce an Expression for Equivalent Capacitance C When Three Capacitors C1, C2 and C3 Connected in Parallel.
- A wire of resistance ‘R’ is cut into ‘n’ equal parts. These parts are then connected in parallel with each other. The equivalent resistance of the combination is:
Concepts [33]
- Introduction to Electric Potential
- Electric Potential: A Quantitative Approach
- Potential Difference
- Work Done in Moving a Charge in an Electric Field
- Acceleration of a Charged Particle Between Two Points in an Electric Field
- Electric Potential Due to a Point Charge
- Potential due to a Group of Point Charges
- Potential Gradient
- Electric Field as Gradient of Electric Potential: Relation between E and V
- Equipotential Surfaces
- Electric Potential Energy of a System of Charges
- Charged Body Between Parallel Plates
- Potential Due to an Electric Dipole
- Work Done in Rotating an Electric Dipole in an Electric Field
- Electric Potential Energy of an Electric Dipole in an Electrostatic Field
- Electrostatics of Conductors
- Free and Bound Charges
- Dielectrics
- Electric Polarisation of Dielectrics
- Capacitance of a Conductor
- Capacitance of an Isolated Spherical Conductor
- Potential Energy of a Charged Conductor
- Redistribution of Charges: Common Potential
- Introduction to a Capacitor
- The Parallel Plate Capacitor
- Expression for Capacitance of a Parallel-Plate Capacitor
- Dependence of the Capacitance of a Capacitor
- Capacitance of a Parallel-Plate Capacitor with Dielectric Slab between Plates
- Combination of Capacitors
- Energy Stored in a Charged Capacitor
- Force between the Plates of a Charged Parallel-Plate Capacitor
- Effect of Dielectric Insertion on a Capacitor: with and Without a Battery
- Variation of Electric Field and Potential Due to a Charged Sphere
