Govt. Exams
Entrance Exams
If isolated (constant charge): U ∝ 1/C, and C increases by K, so U decreases by K. If connected to battery (constant V): U ∝ C, increases by K. Given 'charged' implies isolated.
For zero potential: 8/(x) = 2/(3-x). Solving: 8(3-x) = 2x → 24 = 10x → x = 2.4 m
Potential energy of dipole: U = -p·E = -pE cosθ, where θ is angle between p and E. Minimum at θ = 0
Capacitance of isolated sphere: C = 4πε₀R, where R is radius. Since k = 1/(4πε₀), C = R/k
By Gauss's law, field outside depends only on total enclosed charge Q. E = kQ/r² for r > R₂
Electrostatic force is conservative, meaning work depends only on initial and final positions, not the path taken
For isolated capacitor, charge Q remains constant. U = Q²/(2C) = Q²d/(2ε₀A). Energy is proportional to d, so doubling d doubles energy... correction: U = CV²/2, but Q is constant so U = Q²/(2C) ∝ d, energy increases
By Gauss's law, in the region between spheres, only inner charge +Q contributes. E = kQ/r², directed radially outward
Using Gauss's law for an infinite plane sheet: E = σ/(2ε₀). The field is independent of distance and uniform on both sides
Distance from each vertex to centroid = a/√3. Total potential = 3 × kq/(a/√3) = 3√3kq/a. Correction: V = 3kq√3/a ≈ 3kq/a for approximation