class 12th physics 2025 paper leak 10 que leak

1. Gauss's Theorem and Application to an Infinitely Long Charged Wire

Gauss's Theorem: The total electric flux through a closed surface is equal to $\frac{1}{ε_{0}}$ times the total charge enclosed.

\oint E \cdot d A = \frac{Q_{enc}}{ε_{0}}

Application to an Infinitely Long Charged Wire

For a wire with charge density $λ$ (charge per unit length) at a distance $r$ , using a cylindrical Gaussian surface:

E \cdot (2 π r L) = \frac{λ L}{ε_{0}}

Solving for $E$ :

E = \frac{λ}{2 π ε_{0} r}

📌 The field is radially outward if $λ > 0$ and inward if $λ < 0$ .

2. Potential Energy of a System of Two Point Charges

The electrostatic potential energy of two point charges $q_{1}$ and $q_{2}$ separated by distance $r$ :

U = \frac{1}{4 π ε_{0}} \frac{q_{1} q_{2}}{r}

✅ If $q_{1}$ and $q_{2}$ have the same sign → $U > 0$ (repulsion).
✅ If $q_{1}$ and $q_{2}$ have opposite signs → $U < 0$ (attractions).

3. Kirchhoff's Laws

(i) Kirchhoff's Current Law (KCL):

At any junction, the total incoming current equals the total outgoing current:

\sum I_{in} = \sum I_{out}

(ii) Kirchhoff's Voltage Law (KVL):

The algebraic sum of all potential differences in a closed loop is zero:

\sum V = 0

🔹 Example: In a circuit with resistors and batteries, the sum of EMFs equals the sum of potential drops.

4. Biot-Savart Law and Magnetic Field at the Center of a Circular Loop

Biot-Savart Law: The magnetic field $d B$ due to a small current element $I d l$ at distance $r$ is:

d B

style="vertical-align: inherit;">= μ 0 4 π I d l sin ⁡ θ r 2 dB = \frac{\mu_0}{4\pi} \frac{I \, dl \sin\theta}{r^2}

d B = \frac{μ}{4π ​} \frac{I d l sin}{r ^{2}}

For a current-carrying circular loop of radius $R$ , the field at the center is:

B = \frac{μ_{0} I}{2 R}

📌 The field follows the right-hand thumb rule .

5. Magnetic Field on the Axial Line of a Magnetic Dipole

For a bar magnet or small magnetic dipole the field at an axial distance $r$ from the center:

B = \frac{μ_{0}}{4 π} \frac{2 M}{r^{3}}

where $M = m \cdot 2 l$ is the magnetic dipole moment.

✅ Direction: Along the axis of the dipole.
✅ Behavior: Decrease with $\frac{1}{r^{3}}$ .

6. Faraday's Laws of Electromagnetic Induction

(i) First Law:

A changing magnetic flux induces an EMF in a coil.

(ii) Second Law:

The induced EMF is proportional to the rate of change of flux:

E = - \frac{d Φ_{B}}{d t}

(🔹 Negative sign indicates Lenz's Law , opposing the cause of induction.)

7. Impedance and Resonance in an LCR Circuit

Impedance (Z):

Total opposition to AC in an LCR circuit:

Z = \sqrt{R^{2} + (X_{L} - X_{C})^{2}}

where $X_{L} = ω L$ and $X_{C} = \frac{1}{ω C}$ .

Resonance Condition:

When $X_{L} = X_{C}$ , impedance is minimum and current is maximum:

f_{r} = \frac{1}{2 π \sqrt{L C}}

8. Displacement Current and Maxwell's Equations

🔹 Displacement Current ( $I_{d}$ ) : Accounts for a changing electric field in capacitors.

I_{d} = ε_{0} \frac{d Φ_{E}}{d t}

📌 It completes Ampere's Law , leading to Maxwell's equations .

9. Lens Maker's Formula

For a thin lens with refractive $n$ , radius $R_{1}$ , $R_{2}$ :

\frac{1}{f} = (n - 1) (\frac{1}{R_{1}} - \frac{1}{R_{2}})

✅ Used for convex and concave lenses.

10. Young's Double-Slit Experiment and Fringe Width

For two slits separated by distance $d$ , wavelength $λ$ , and screen distance $D$ :

Fringe width w = \frac{λ D}{d}

✅ Bright fringes : Constructive interference.
✅ Dark fringes : Destructive interference.

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