University:
Santa Fe College
Course:
ETS 2527C | Electromechanical Components and Mechanisms
Academic year:

2024
Views:

238

Pages:

6
Author:

Jabin F.

Title: # Dipole moment of a continuous charge system Formulas: (P = \int_{C} \lambda \vec{r} dl) (P = \int_{S} \sigma \vec{r} ds) (P = \int_{V} \rho \vec{r} d\tau) Statement: "If charge is distributed uniformly over any shape, its dipole moment will be zero." Conditions: (λ = λ_0) (σ = σ_0) (ρ = ρ_0) Examples: (ρ = ρ_0 \gamma) (dipole moment will be zero) (ρ = ρ_0 \cos θ) (dipole moment will be nonzero) (φ = ρ_0 \sin θ) (dipole moment will be zero) Note: "Gauss law not applicable for Dipole" Header: Ques. 1 Diagram: An equilateral triangle ABC with sides of length 'a'. Charges -2q, q, and q are placed at vertices A, B, and C, respectively. Text: "Charges -2q, q, and q are placed on the vertices of an equilateral triangle ABC of sides of length a, as shown in figure." "The dipole moment of this configuration of charges, irrespective of the choice of origin is" "Let origin at A" (P_A = q(0\hat{x} + 0\hat{y}) + q(a\hat{x} + 0\hat{y}) + (-2q)(\frac{a}{2}\hat{x} + \frac{\sqrt{3}a}{2}\hat{y})) (P_A = q\hat{x}a - q\hat{x}a - \sqrt{3}a\hat{y}q) (P_A = -\sqrt{3}q\hat{y}a) "equivalent" (P_0 = \sqrt{3}q\hat{y}a) "If" (V = p_0s_0) (P = \sqrt{3}q\hat{y}a) Header: Ques. 2 Diagram: A square with charges -2q, q, q, and q placed at its vertices. The origin is chosen at one of the vertices. Text: "Here Dipole moment depends upon choice of origin." "Here not mention the origin so we choose at O(0,0)." (P_0 = (+2q)(0) + 2q(a\hat{x}) + 2q(a\hat{y}) + (-2q)(a\hat{x} + a\hat{y})) (P_0 = -2qa\hat{x} - 2qa\hat{y}) (|P_0| = \sqrt{2}qa) "At large distances (r>>a)" "In the multipole expansion of potential V, the dominant term due to" "(i) Monopole: (V_{monopole} \propto \frac{1}{r}), (E_{monopole} \propto \frac{1}{r^2})" "(ii) Dipole: (V_{dipole} \propto \frac{1}{r^2}), (E_{dipole} \propto \frac{1}{r^3})" "(iii) Quadrupole: (V_{quadrupole} \propto \frac{1}{r^3}), (E_{quadrupole} \propto \frac{1}{r^4})" "(iv) Octopole: (V_{octopole} \propto \frac{1}{r^4}), (E_{octopole} \propto \frac{1}{r^5})" "If monopole is non-zero, then dominant is monopole." "If monopole is zero, then we calculate dipole."