Electric Charges and Fields - Class 12 Bihar Board Physics

Electric Charges and Fields

Electrostatics: Definition and Scope

Electrostatics is the branch of physics that deals with electric charges at rest. It focuses on the properties, behavior, and effects of stationary charges and the fields they produce. The main topics under electrostatics include electric charge, electric field, electric flux, Gauss's theorem, electric potential, and capacitance.

• Electrostatics = "Electron" (charge) + "Statics" (rest)
• Exam weightage: 8–10 marks from this chapter; Chapters 1 and 2 together contribute about 21 marks.

Electric Charge: Nature, Types, and Units

Electric charge is a fundamental property of matter that causes it…

Electric Charges and Fields

Electrostatics: Definition and Scope

Electrostatics is the branch of physics that deals with electric charges at rest. It focuses on the properties, behavior, and effects of stationary charges and the fields they produce. The main topics under electrostatics include electric charge, electric field, electric flux, Gauss's theorem, electric potential, and capacitance.

• Electrostatics = "Electron" (charge) + "Statics" (rest)
• Exam weightage: 8–10 marks from this chapter; Chapters 1 and 2 together contribute about 21 marks.

Electric Charge: Nature, Types, and Units

Electric charge is a fundamental property of matter that causes it to experience a force when placed in an electric field. It is denoted by $Q$ or $q$ and can be positive (proton) or negative (electron). The SI unit of charge is the coulomb (C).

• Types of Charge: Positive (proton), Negative (electron)
• Charge on 1 proton: $+1.6 \times 10^{-19}$ C
• Charge on 1 electron: $-1.6 \times 10^{-19}$ C
• Magnitude is same for electron and proton; sign indicates nature
• Formula: $q = It$ (current × time)
• SI Units: Coulomb (C), Ampere-hour (A·h), Ampere-second (A·s)
• CGS Unit: Statcoulomb (esu)
• Conversions:
  • $1$ statcoulomb $= \frac{1}{3} \times 10^{-9}$ C
  • $1$ C $= 3 \times 10^{9}$ statcoulomb
• Dimension: $[A][T]$
• Specific charge of electron: $1.76 \times 10^{11}$ C/kg

Charging and Discharging of Bodies

A body becomes charged by gaining or losing electrons. Removal of electrons results in a positive charge, while gaining electrons results in a negative charge. The mass of the body decreases when electrons are removed and increases when electrons are gained.

• Neutral body: Number of electrons = Number of protons
• Positive charge: Electrons removed ($N_{proton} > N_{electron}$)
• Negative charge: Electrons gained ($N_{electron} > N_{proton}$)
• Effect on mass:
  • Positive charge (electron removal): Mass decreases
  • Negative charge (electron gain): Mass increases
• Mass of electron: $9.11 \times 10^{-31}$ kg
• Mass of proton: $1.67 \times 10^{-27}$ kg

Conservation and Quantization of Charge

Charge is conserved; it can neither be created nor destroyed, only transferred from one body to another. Charge is also quantized, meaning it exists in discrete integral multiples of the elementary charge ($e$).

• Conservation Law: Total charge before and after any process remains the same.
• Quantization Formula: $q = n e$
  • $n$ = integer (never fractional or decimal)
  • $e = 1.6 \times 10^{-19}$ C
• Fractional values of $n$ are not possible.
• Charge on any body is always an integral multiple of $e$.

Coulomb’s Law: Statement, Formula, and Vector Form

Coulomb’s Law gives the force between two point charges. The force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them.

• Scalar Form: $F = k \frac{q_1 q_2}{r^2}$
  • $k = \frac{1}{4\pi \epsilon_0}$
  • $k = 9 \times 10^9$ N·m$^2$/C$^2$
• $\epsilon_0$ (permittivity of free space): $8.85 \times 10^{-12}$ C$^2$/N·m$^2$
• Vector Form:
  • If denominator is $|\vec{r}|^3$, numerator is $\vec{r}$ (vector)
  • If denominator is $|\vec{r}|^2$, numerator is $\hat{r}$ (unit vector)
• Coulomb’s Law follows Newton’s Third Law: Action and reaction forces are equal and opposite.
• Limitations:
  • Applicable only for point charges at rest
  • Only two charges at a time

Electric Field and Electric Field Intensity

The electric field is the region around a charge where it exerts a force on other charges. Electric field intensity at a point is the force experienced by a unit positive test charge placed at that point.

• Definition: $E = \frac{F}{q_0}$
• Unit: N/C or V/m
• Dimension: $M L T^{-3} A^{-1}$
• For a point charge: $E = k \frac{q}{r^2}$
• Test charge is always positive to determine the nature of the source charge.
• Direction:
  • For positive source charge: Outward
  • For negative source charge: Inward

Electric Field Due to Multiple Charges (Superposition Principle)

The net electric field at a point due to multiple charges is the vector sum of the fields due to individual charges.

• $E_{total} = E_1 + E_2 + E_3 + \ldots + E_n$
• For each charge: $E_i = k \frac{q_i}{r_i^2}$

Electric Field Lines and Their Properties

Electric field lines are imaginary lines used to represent the direction and strength of the electric field. They start from positive charges and end on negative charges.

• Properties:
  • Originate from positive, terminate at negative charges
  • Tangent at any point gives the direction of the field
  • Never intersect each other
  • Density of lines indicates field strength
  • Perpendicular to the surface of conductors
  • Electric potential decreases in the direction of the field

Electric Dipole and Dipole Moment

An electric dipole consists of two equal and opposite charges separated by a small distance. The dipole moment ($\vec{p}$) measures the strength of the dipole.

• Dipole moment: $\vec{p} = q \cdot 2l$ (where $2l$ is the separation)
• Unit: C·m
• Dimension: $A T L$
• Total charge on dipole: Zero

Electric Field Due to a Dipole

The electric field due to a dipole is calculated at two main positions: axial (on the axis) and equatorial (perpendicular bisector).

• Axial Point (Short Dipole):$$E_{axial} = \frac{k \cdot 2p}{r^3}$$
• Equatorial Point (Short Dipole):$$E_{equatorial} = \frac{k \cdot p}{r^3}$$
• Ratio: $E_{axial} : E_{equatorial} = 2:1$

Torque on a Dipole in Uniform Electric Field

A dipole placed in a uniform electric field experiences a torque that tends to align it with the field.

• Torque: $\tau = p E \sin\theta$
• Vector Form: $\vec{\tau} = \vec{p} \times \vec{E}$
• Equilibrium:
  • Stable equilibrium: Angle between $\vec{p}$ and $\vec{E}$ is $0^\circ$

Continuous Charge Distributions

Charge can be distributed over length, area, or volume.

• Linear charge density ($\lambda$): $\lambda = \frac{Q}{l}$, unit: C/m, dimension: $A T L^{-1}$
• Surface charge density ($\sigma$): $\sigma = \frac{Q}{A}$, unit: C/m$^2$, dimension: $A T L^{-2}$
• Volume charge density ($\rho$): $\rho = \frac{Q}{V}$, unit: C/m$^3$, dimension: $A T L^{-3}$

Electric Flux

Electric flux measures the number of electric field lines passing normally through a given area.

• Definition: $\Phi = \vec{E} \cdot \vec{A} = EA \cos\theta$
• Unit: N·m$^2$/C or V·m
• Dimension: $M L^3 T^{-3} A^{-1}$
• Area vector is always perpendicular to the surface.

Gauss’s Theorem

Gauss’s theorem relates the total electric flux through a closed surface to the charge enclosed by that surface.

• Statement: Total electric flux through a closed surface is $1/\epsilon_0$ times the total charge enclosed.
• Formula: $\Phi = \frac{q_{encl}}{\epsilon_0}$
• Applicable only for closed surfaces.

Applications and Derivations Using Gauss’s Theorem

Gauss’s theorem can be used to derive Coulomb’s law and to calculate the electric field for various charge distributions.

• Derivation of Coulomb’s Law:
  • For a spherical Gaussian surface of radius $r$ enclosing charge $q$:$$\Phi = E \cdot 4\pi r^2 = \frac{q}{\epsilon_0}$$$$E = \frac{1}{4\pi \epsilon_0} \frac{q}{r^2}$$
• For infinite long wire (linear charge density $\lambda$):$$E = \frac{\lambda}{2\pi \epsilon_0 r}$$
• For infinite charged sheet (surface charge density $\sigma$):$$E = \frac{\sigma}{2\epsilon_0}$$

Special Cases and Exam-Oriented Points

Certain results and properties are frequently tested in exams.

• Electric field inside a charged spherical shell: Zero
• Electric field on the surface of a charged sphere: $E = k \frac{Q}{R^2}$
• Electric field due to a circular loop at a point on its axis:$$E = \frac{1}{4\pi \epsilon_0} \frac{q x}{(r^2 + x^2)^{3/2}}$$
  • $r$ = radius of loop, $x$ = distance from center to point $P$
• Electric field inside a conductor: Zero
• Electric field at the surface of a conductor: Perpendicular to the surface

Key Units and Dimensions Table

• Relative permittivity ($\epsilon_r$): $\epsilon_r = \frac{\epsilon}{\epsilon_0}$, unitless and dimensionless, also called dielectric constant.
• Permittivity of free space ($\epsilon_0$): $8.85 \times 10^{-12}$ C$^2$/N·m$^2$
• Permittivity of air: 1

Exam-Focused MCQ and Short Answer Traps

• Charge on electron: $-1.6 \times 10^{-19}$ C

• 1 C = $3 \times 10^9$ statcoulomb

• Specific charge of electron: $1.76 \times 10^{11}$ C/kg

• Dimension of charge: $A T$

• Electric field unit: N/C or V/m

• Electric flux unit: N·m$^2$/C or V·m

• Electric field at conductor surface: Perpendicular

• Electric field inside conductor: Zero

• Electric field lines never cross each other

• Electric field lines start from positive and end at negative charges

• Electric field intensity at surface of conductor: Perpendicular to surface

• For all calculations, use only the values and formulas explicitly provided above.