Electric Charges and Fields (PDF)

# Electric Charges and Fields ## Charge | Type | Electric Field (EF) | Magnetic Field (MF) | Electromagnetic Wave (EM wave) | |---|---|---|---| | Rest | ✓ | X | X | | V = const | ✓ | ✓ | X | | Accelerated | ✓ | ✓ | ✓ | ## Charge! * **Scaler** * **SI unit:** Coulomb (C) * **C.G.S unit:** e.s.u (elect...

# Electric Charges and Fields ## Charge | Type | Electric Field (EF) | Magnetic Field (MF) | Electromagnetic Wave (EM wave) | |---|---|---|---| | Rest | ✓ | X | X | | V = const | ✓ | ✓ | X | | Accelerated | ✓ | ✓ | ✓ | ## Charge! * **Scaler** * **SI unit:** Coulomb (C) * **C.G.S unit:** e.s.u (electrostatic unit) * **Practical unit:** e.m.u (electromagnetic unit) = 1frankline * **1C = 3 x 10^9 e.s.u/frankline ** * **Smallest unit:** e.s.u/frankline * **Largest unit:** Faraday = 96500C ## Types of Charges * **Positive charge:** deficiency of electrons * **Negative charge:** Excessness of electrons * **Neutral body:** no of electron = no of proton * **eg** Neutron ## Properties of Charge * **Scaler, conserved, Quantized** * **Q = ne**, n = integer * **Similar charges:** Generally repel, may attract * **Opposite charge:** Must attract, but mass can exist without charge but charge can't exist without mass. * **Charge follow simple scalar addition.** * **Charge is transferable.** * **Invariant:** Doesn't depend upon speed. ## Specific Charge | Particle | e/m | |---|---| | Electron | e/me | | Proton | e/mp | | Deutron | e/2mp | | Alpha particle | e/2mp | * **charge same** * **e = se** * **m = mt** * **charge same** * **e = se** * **m = mb** ## Conductor/Insulator If excess charge given to a: 1. **Hollow conducting sphere:** Remain on the surface. 2. **Solid conducting sphere:** 3. **Hollow insulating sphere:** Remain through out the volume. 4. **Solid insulating sphere:** ## Charging of Body 1. **By Friction:** Only for Insulator. When rubbing, equal and opposite nature of charge created. 2. **By conduction:** * **New charge = UskaR[Total]/Total R** * **Q₁ = R1[Q₁+Q₂] / R1+R2** 3. **By Induction:** * **Conductor:** Qinduced = Qinducing * **Dielectric:** Qinduced < Qinducing ## Coulomb's Law * **F12 = F21** (Action reaction pair). * **Central, Conservative (long range)** * **Follows Inverse square law.** * **Depends on medium** * **F=KQ₁Q₂/r²** * **K=1/4πε₀=9x10^9 Nm²/C²** * **Kmedium=1/ε = 1/ε₀εr = 9x10^9 Nm²/C²/εr** * **Fmed = Fair/εr** * **ε₀ = 8.85×10^-12 C²/Nm²** (space) * **εr = Relative permittivity** ## Superposition Theorem * **Force blw any two charges does not depend on presence of the other charges.** * **Fnet = F₁ + F₂ + F₃....** ## Position of 3rd Charge * Force on that charge will be zero. * **Q₁=+Q, Q₂=+ng** **Case I:** Like charge * **x = ar/(√n+1)** * **From smaller charge** **Case II:** Unlike charge * **x = ar/(√n-1)** ## For System to be in Equilibrium * **x = a/(√n+1)** ** Q = -nq/(√n+1)²** * **If symmetrical then Fnet = 0** * **Fnet = KQL/(a√2)²** ## Pendulum Problem * **T = 2π√l/g** * **tanθ = Fe/mg = (KQ₁Q₂/x²)/mg** * **tanθ₁ = KQ₁Q₂/x²m₁g** * **tanθ₂ = KQ₁Q₂/x²m₂g** * **Ratio of θ₁ & θ₂ does not depend on charges & depends only on mass.** ## Coulomb's Law in Vector form * **F21 = KQ₁Q₂r21/│r21│³ = KQ₁Q₂r21/│r21│²** * **F21 = KQ₂Q₁r12/│r12│³ = KQ₂Q₁r12/│r12│²** # Electric Field ## Definition * Electrostatic force experienced by +1C charge. * **E = F/q = N/C** * **E = +ve** on **+ve** * **E = -ve** on **-ve** ## Electric field on point charge * **+Q, +q,** very small * **E = KQ/r²** ## Electric field due to a ring * E = KQx/(R²+x²)³/² * **at x=±R/√2, Emax ≈ 2KQ/3√3R²** * **x>>>>R, E = KQx/R³ ≈ KQ/R²** * **x = 0; E = KQ/R²** * **E = KQx/(R²+x²)³/²** ## Field due to circular arc * **E = Kqxsin(θ/2)/R²** * **E = 2KQsin(θ/2)/R²** ## Linear Charge density **For half-ring:** * **E = 2Kλ/R** ## Electric field lines * The imaginary lines, tangent to which gives the direction of electric field. * No. of field lines passing through per unit area gives the strength of field. # E-field due to combination of sheet * **A, B, +σ** * **C, -σ** * **Ε = 0** * **Ε = +σ/ε₀ ** * **Ε = -σ/ε₀** * **Ε = 2σ/ε₀** ### Thick Spherical Shells * **Ka/r² (r < a)** * ** KQ/r² (a < r < b)** * **KQ/r² (r > b)** * **Kq/r² ( a < r < b) * **Kq/r² ( r > b)** # Charge distribution on concentric conducting sphere The sum of all charge will reflect on the outer surface of the outer sphere. * **Hoc: 10 - 6 + 2 = 6 + 6** # Charge distribution by induction on conducting parallel plates * **Charge on outer plate = total charge** * **Charge on facing sides are equal and opposite** * **Total charge conserved on each plate.** # E-field of a charge * **E-field line do not along the path followed by unit +ve charge.** * **Starts from the charge, ends at -ve charge.** * **The tangent at any point on the electric field line gives the direction of field.** * **Radially outward from the +ve charge and radially inward from the -ve charge.** * **Do not intersect two lines.** * **Magnitude of charge is proportional to no. of field lines.** * **Electric field line density** * **Always perpendicular to the surface inside the conductor.** * **Inside the conductor, E = 0** * **Sharp sharp turn not possible, always continuous at all points.** * **Conservative field and never forms a closed loop.** * **uniform E and non-uniform E.** # Motion of Charge Particle in uniform Electric Field **1. Charges is released** * V = 0 * F = qE * a = qE/m * V = qEt/m * KE = p²/2m = q²E²t²/2m * P = qEt * S = 1/2qEt²/2m = qEt²/2m * Distance **2. Charge is projected, parallel to electric field.** * **V = V₀t + qEt/m** * **S = V₀t + 1/2qEt²/m** **3. Charge is projected perpendicular to E** * **tan θ = qEd/mv²** * **S = qEd²/2mv²** # Electric dipole * Two equal & opposite charges placed at a small distance. * **Vector, unit→ Cm, [A'T'L']** * **Directed from -q to +q** * **Ideal dipole: I very small dipole** * **Eaxial = (4kqr/ (a²-r²)²) = 2kpr/(a²- r²)²** * **Line dipole is small** * **Eaxial = 2KP/r³** * **Eequitorial = (2KQa/ (a²+r²)³/²) = KP /(a²+r²)³/²** * **Small dipole (r << a)** * **Eequitorial = -KP/r³** * **E = 2kPcOSO/r3** * **Enet = KP √(1+3Cos²θ)/r³** * **tanθ = tanJ2** * **Angle blw E & P = θtanJ2** * **Direction of E = tanJ2** # Force blw two point charge Fα1/r² # Force blw dipole and point charge Fα1/r^4 # Force blw two dipoles Fα1/? # Electric flux * Counting of no. of E.f lines passing though given area. * **Scalar, Unit - Nm²/c** * **It gives the idea of electrostatic energy passing through a given area.** * **Uniform E → Φ = E.A = EAcosθ** * **Variable E→ Φ = ∫E.dA** #Flux for various objects * **1D object** → Not defined * **2D object → Φ passing = EA** * **3D object:** * **Φincoming = -ve, Φout = +ve** * **Total Φ = Φin + Φout** * **Uniform E, Φ = 0** ## Angle blw surface and E field * **Φ = EAcos(90-θ)** * **Angle b/w Area and surface is always 90°** # Gauss Law * **The total flux passing through any closed surface = 2/ε₀ x enclosed charge** * **Φ = ∫E.dA = ∮Lin = 2/ε₀ x Σenclosed charge** * **Φ = ES = 2/ε₀ x Σtotal charge** ## Important Points * **Gauss law is true for an closed surface, no matter what its shape or size. Only depends on charge enclosed.** * **Does not depend on location of enclosed charge.** * **Gauss law is always valid, but it may not be useful to calculate E field.** * **Useful to calculate E field for symmetrical charge distribution.** # Applying Gauss law to calculate E-field **1. Electric field due to a point charge.** * **Q, R, E** * **E = Kq/R²** **2. Infinite line charge** * **λ, R, E** * **E = 2Kλ/r** * **Eα1/r** **3. Infinite non-conducting charged sheet** * **σ, x, E** * **E = σ/2ε₀** **4. Infinite conducting charged sheet** * **σ, x, E** * **E = σ/ε₀** **5. Hollow solid conducting sphere, non-conducting hollow sphere** * **Inside (r<R), E = 0** * **r=R, Ecentre = 0** * **Esurface = KQ/R², Gout = KQ/r², same as point charge outside.** **6. Solid non-conducting sphere** * **Uniformly charged sphere** * **Q, R, σ, x** * **Gout (r>R) = KQ/r²** * **Gin(r<R) = KQ/r² = σr/ε₀** * **Ecent = kQ/R² = σ/ε₀** **7. E.F inside cavity of non-conducting solid sphere** * **E p = E complete - E cavity** * **Einside = KQ/r³ = σr/ε₀**