Gauss_Law_Investigatory_Project. FOR 12 STUDENTS

GAUSS LAW EXPLANATION

1. Gauss's Law - Investigatory Project Class XII Physics Name: __________ Roll No: _______ School: _______ Guide: _______ Date: _______

2. Aim To verify Gauss's Law by showing that the electric flux through a closed spherical surface is constant for different radii enclosing the same charge, and that flux is proportional to the enclosed charge.

3. Theory / Background Gauss's Law: The net electric flux Φ through any closed surface = Q_enclosed / ε₀. Mathematically: Φ = ∮ E · dA = Q_enclosed / ε₀ For a point charge q at the center of a spherical surface of radius r, E = kq/r² and dA = r² sinθ dθ dφ (constant magnitude r² for sphere). Hence Φ = E (4πr²) = (kq/r²) (4πr²) = 4πk q = q/ε₀, independent of r.

4. Apparatus 1. Small conducting sphere (metallic ball) - simulated as a point charge in theory. 2. High-voltage supply / charging apparatus (for real experiment). 3. Electrometer / Field sensor (in practical labs). 4. Insulating stand, connecting wires, and measuring scale (for alignment). 5. Vernier calipers, ruler, stopwatch (for precision). Note: In this investigatory project we demonstrate and verify Gauss's Law using simulation and schematic diagrams suitable for Class XII.

5. Apparatus & Schematic Diagrams

6. Experimental Procedure 1. Place the conducting sphere at the center of an insulating stand. 2. Charge the sphere with a known charge q using the charging apparatus (or assume q for simulation). 3. Choose several concentric spherical Gaussian surfaces (different radii) centered on the charge. 4. Measure the electric field E at the surface for each radius using an electrometer or field sensor (or compute theoretical E). 5. Calculate flux Φ = E × (surface area) = E × 4πr² for each radius. 6. Repeat measurements for different values of q and record values of Φ. 7. Plot Φ vs r and Φ vs q to verify independence from r and proportionality to q.

7. Simulated Experimental Data Simulated Data (Fixed charge q = 5.0×10⁻⁹ C) Radius = 0.050 m | E = 1.798e+04 | Φ_calc = 5.647e+02 | Φ_meas = 5.831e+02 Radius = 0.080 m | E = 7.022e+03 | Φ_calc = 5.647e+02 | Φ_meas = 5.578e+02 Radius = 0.120 m | E = 3.121e+03 | Φ_calc = 5.647e+02 | Φ_meas = 5.587e+02 Radius = 0.200 m | E = 1.123e+03 | Φ_calc = 5.647e+02 | Φ_meas = 5.526e+02 Radius = 0.350 m | E = 3.668e+02 | Φ_calc = 5.647e+02 | Φ_meas = 5.745e+02

8. Graph: Flux vs Radius

9. Graph: Flux vs Enclosed Charge

10. Observations • The measured flux values remain approximately constant for different radii enclosing the same charge. • The flux is directly proportional to the enclosed charge.

11. Precautions • Ensure the charge is centered within Gaussian surfaces. • Use well-calibrated field sensors/electrometers. • Minimize external electric fields and shielding effects. • Use insulating stands to avoid charge leakage.

12. Conclusion The experiment (simulated) verifies Gauss's Law: electric flux through a closed surface depends only on the enclosed charge and is independent of the size of the Gaussian surface. The simulated data and graphs support the theory within expected experimental errors.

13. References • Halliday, Resnick & Walker - Fundamentals of Physics • Practical Physics (University lab manuals) • Class XII Physics NCERT - Electrostatics chapters