MP Board Class 12 Physics: Quick Revision Notes & Formula Sheet

Chapter 1: Electric Charges and Fields (विद्युत आवेश तथा क्षेत्र)

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1. Fundamental Properties of Electric Charge (विद्युत आवेश)

• Electric Charge ($q$ or $Q$): Intrinsic property of matter due to which it experiences and produces electrical and magnetic effects.
• SI Unit: Coulomb ($\text{C}$) | Dimension: $[A T]$
• Quantization of Charge: Charge exists in discrete packets. $$q = \pm ne$$ (where $n = 1, 2, 3...$ and $e = 1.6 \times 10^{-19}\text{ C}$)
• Conservation of Charge: Total charge of an isolated system remains constant.
• Additive Nature: $Q_{total} = q_1 + q_2 + q_3 + \dots + q_n$

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2. Coulomb's Law in Electrostatics (कूलाम का नियम)

The electrostatic force of attraction or repulsion between two point charges is directly proportional to the product of their magnitude and inversely proportional to the square of the distance between them.

$$F = \frac{1}{4\pi\varepsilon_0} \cdot \frac{|q_1 q_2|}{r^2}$$

• In Vacuum / Air: $$k = \frac{1}{4\pi\varepsilon_0} \approx 9 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2$$
• Permittivity of Free Space ($\varepsilon_0$): $$\varepsilon_0 = 8.854 \times 10^{-12} \text{ C}^2/(\text{N}\cdot\text{m}^2) \quad \text{or} \text{ F/m}$$ $$\text{Dimensional Formula of } \varepsilon_0 = [M^{-1} L^{-3} T^4 A^2]$$
• In a Dielectric Medium: $$F_m = \frac{F_{air}}{K} = \frac{1}{4\pi\varepsilon_0 K} \cdot \frac{|q_1 q_2|}{r^2}$$ (where $K = \varepsilon_r$ is the Dielectric Constant or Relative Permittivity, $K \ge 1$)

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3. Charge Distributions (आवेश वितरण)

Type	Charge Density Symbol	Formula	SI Unit
Linear Charge Density (रेखीय आवेश घनत्व)	$\lambda$	$\lambda = \frac{q}{L}$	$\text{C/m}$
Surface Charge Density (पृष्ठीय आवेश घनत्व)	$\sigma$	$\sigma = \frac{q}{A}$	$\text{C/m}^2$
Volume Charge Density (आयतन आवेश घनत्व)	$\rho$	$\rho = \frac{q}{V}$	$\text{C/m}^3$

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4. Electric Field (विद्युत क्षेत्र)

• Electric Field Intensity ($\vec{E}$): Force experienced per unit positive test charge. $$\vec{E} = \frac{\vec{F}}{q_0}$$
• SI Unit: $\text{N/C}$ or $\text{V/m}$ | Dimension: $[M L T^{-3} A^{-1}]$
• Electric Field due to a Point Charge $q$ at distance $r$: $$E = \frac{1}{4\pi\varepsilon_0} \cdot \frac{q}{r^2}$$

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5. Electric Dipole (विद्युत द्विध्रुव)

• Electric Dipole Moment ($\vec{p}$): Vector quantity directed from Negative charge ($-q$) to Positive charge ($+q$). $$p = q \times 2a$$ (where $2a$ is the dipole length)
• SI Unit: $\text{Coulomb-meter } (\text{C}\cdot\text{m})$ | Dimension: $[L T A]$

Key Field Formulas for Electric Dipole:

1. Axial Position (End-on Position / अक्षीय स्थिति): $$E_{axial} = \frac{1}{4\pi\varepsilon_0} \cdot \frac{2pr}{(r^2 - a^2)^2}$$ For a short dipole ($r \gg a$): $$E_{axial} = \frac{1}{4\pi\varepsilon_0} \cdot \frac{2p}{r^3}$$

2. Equatorial Position (Broadside-on Position / निरक्षीय स्थिति): $$E_{eq} = \frac{1}{4\pi\varepsilon_0} \cdot \frac{p}{(r^2 + a^2)^{3/2}}$$ For a short dipole ($r \gg a$): $$E_{eq} = \frac{1}{4\pi\varepsilon_0} \cdot \frac{p}{r^3}$$

3. Relation for Short Dipole: $$E_{axial} = 2 \times E_{eq}$$

4. Torque ($\vec{\tau}$) on Dipole in Uniform Field: $$\vec{\tau} = \vec{p} \times \vec{E} \implies \tau = pE \sin\theta$$

   • Maximum Torque: $\tau_{max} = pE$ (when $\theta = 90^\circ$)
   • Stable Equilibrium: $\theta = 0^\circ$ ($\tau = 0$)
   • Unstable Equilibrium: $\theta = 180^\circ$ ($\tau = 0$)

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6. Electric Flux (विद्युत फ्लक्स)

Total number of electric field lines passing normally through a surface.

$$\Phi_E = \vec{E} \cdot \vec{A} = E A \cos\theta$$

• Scalar Quantity
• SI Unit: $\text{N}\cdot\text{m}^2/\text{C}$ or $\text{Volt-meter } (\text{V}\cdot\text{m})$
• Dimensional Formula: $[M L^3 T^{-3} A^{-1}]$

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7. Gauss's Law (गॉस का नियम)

The total electric flux passing through any closed Gaussian surface is equal to $\frac{1}{\varepsilon_0}$ times the total charge enclosed by that surface.

$$\Phi_E = \oint \vec{E} \cdot d\vec{A} = \frac{q_{enclosed}}{\varepsilon_0}$$

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8. Applications of Gauss's Law (गॉस के नियम के अनुप्रयोग)

1. Infinitely Long Straight Uniformly Charged Wire: $$E = \frac{\lambda}{2\pi\varepsilon_0 r}$$

2. Infinite Uniformly Charged Thin Plane Sheet: $$E = \frac{\sigma}{2\varepsilon_0}$$ (Field is independent of distance $r$ from the sheet)

3. Uniformly Charged Thin Spherical Shell (Radius $R$, Total Charge $Q$):

   • Outside Shell ($r > R$): $E = \frac{1}{4\pi\varepsilon_0} \cdot \frac{Q}{r^2}$
   • On Surface ($r = R$): $E = \frac{1}{4\pi\varepsilon_0} \cdot \frac{Q}{R^2}$
   • Inside Shell ($r < R$): $E = 0$ (Since charge enclosed inside a conductor shell is zero)

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9. Quick Memory Table for MP Board Objective Questions

Physical Quantity	Symbol	Formula	SI Unit	Dimension
Electric Charge	$q$	$q = ne$	$\text{Coulomb (C)}$	$[A T]$
Electric Field	$E$	$E = F/q$	$\text{N/C}$ or $\text{V/m}$	$[M L T^{-3} A^{-1}]$
Dipole Moment	$p$	$p = q \cdot 2a$	$\text{C}\cdot\text{m}$	$[L T A]$
Electric Flux	$\Phi_E$	$\Phi_E = E A \cos\theta$	$\text{N}\cdot\text{m}^2/\text{C}$	$[M L^3 T^{-3} A^{-1}]$
Permittivity of Free Space	$\varepsilon_0$	$\varepsilon_0 = \frac{q_1 q_2}{4\pi F r^2}$	$\text{C}^2/(\text{N}\cdot\text{m}^2)$	$[M^{-1} L^{-3} T^4 A^2]$

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💡 MP Board Exam Tips

• 2 Marks Question: Frequently asked definitions: Quantization of Charge, Electric Field Line properties, Electric Flux definition with units.
• 3 / 4 Marks Derivations:
  1. Derivation of Electric Field in Axial and Equatorial positions for an Electric Dipole.
  2. State and prove Gauss's Law.
  3. Electric Field due to an infinitely long straight charged wire using Gauss's Law.