B2, General Physics Experiment II E Fall Semester, 2021 Name: Team No. : Department: Date : Student ID: Lecturer’s Signature : Introduction Goals • Understand the relationship between equipotentials and electric fields. • Understand derivatives with discrete algebra. Theoretical Backgrounds 1. Electric Field and Electrostatic Potential (a) The electric field E is defined by E = F q 0 , where F is the electrostatic force applied to a test charge q 0 . (b) The electrostatic potential V at x is defined by V (x)= V (x 0 ) - Z x x 0 E · dx, where V (x 0 ) is the potential at a reference point x 0 . (c) The electric field can be computed from the electrostatic potential as E = -∇V. 2. Electric Field Line (a) An electric field line is an imaginary curve that follows the electric field. (b) If we follow the electric field line, the electrostatic potential decreases. (c) The density of electric field lines is proportional to the strength of the electric field. 3. Equipotential Surface (a) An equipotential surface is a set of points with the same electrostatic potential. (b) The electric field line is normal to the equipotential surface. 4. Electric field and Conductor (a) Inside a conductor the electric field vanishes. (b) Just outside a conductor the electric field is normal to the conducting surface. 5. Computing Derivatives with Discrete Algebra The derivative of a differentiable function f (x) is defined by df (x) dx = lim Δ→0 f (x + Δ) - f (x) Δ . (a) The derivative f 0 (x) exists if the limits exist in both (forward and backward) directions and they are equal. f 0 + (x) ≡ lim Δ→0 + Δ + f (x) Δ , f 0 - (x) ≡ lim Δ→0 - Δ - f (x) Δ . Here, Δ + f (x) [Δ - f (x)] is the forward (backward) difference Δ + f (x) ≡ f (x + Δ) - f (x), Δ - f (x) ≡ f (x) - f (x - Δ). (b) If it is impossible to take the limit Δ → 0, then we need to develop a method to estimate the derivative with a systematic error control. (c) By making use of Taylor’s expansion, we find that Δ + f (x)/Δ has an error of order Δ or higher: Δ + f (x) Δ - f 0 (x) = f (x + Δ) - f (x) Δ - f 0 (x) = 1 2 f 00 (x)Δ + ∞ X k=3 f (k) (x) Δ k-1 k! , where f (k) (x) is the kth-order derivative of f (x). © 2021 KPOPE All rights reserved. Korea University Page 1 of 5