- 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 [22]
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).
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.
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’.
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: 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.
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: Semiconductors
Substances whose resistance to the movement of charges is intermediate between conductors and insulators, are called semiconductors.
Definition: Conductors
Conductors are those through which electric charge can easily flow. Metals, human body, earth, mercury and electrolytes are conductors of electricity.
OR
The material through which electric charge can flow easily is called a conductor.
Definition: Insulators
Those substances in which electric charge cannot flow are called ‘insulators' (or dielectrics). Glass, hard-rubber, plastics and dry wood are insulators. Insulators have practically no free electrons.
OR
The material in which electrons are tightly bound to the nucleus and thus not available for conductance is called an insulator.
OR
Substances which offer high resistance to the passage of electricity and do not allow electricity to pass through them easily, are called insulators.
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: 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: 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.
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.
Formulae [15]
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: 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: Electric Potential at a Point
V = \[\frac {W}{Q}\]
or
W = QV
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 Energy of Two Point Charges
U = \[\frac{1}{4\pi\varepsilon_0}\cdot\frac{q_1q_2}{r_{12}}\]
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: 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: 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: 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: 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: 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: 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: 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)
Formula: Polarisation Vector (P)
Defined as dipole moment per unit volume:
\[P=\frac{\text{dipole moment}}{\mathrm{volume}}=np\]
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 [30]
- Electric Potential
- Potential and Potential Difference
- Electron-volt or eV
- 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
- Electric Potential Energy of a System of Charges
- Equipotential Surfaces
- 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
- Conductors and Insulators
- Free Charges and Bound Charges Inside a Conductor
- Capacitance of a Conductor
- Capacitance of an Isolated Spherical Conductor
- Potential Energy of a Charged Conductor
- Redistribution of Charges: Common Potential
- Capacitors
- Capacitance of a 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
- Induced Charges in a Dielectric Slab in a Capacitor
- Dielectrics
- Electric Polarisation of Matter
- Effect of Introducing a Dielectric between the Plates of a Charged Capacitor
