Download pdf - B2, General Physics Experiment II E Fall Semester, 2021

B2, General Physics Experiment II E Fall Semester, 2021

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Introduction

Goals

• Understand the relationship betweenequipotentials 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

q0,

where F is the electrostatic force applied to atest charge q0.

(b) The electrostatic potential V at x is definedby

V (x) = V (x0)−∫ x

x0

E · dx,

where V (x0) is the potential at a referencepoint x0.

(c) The electric field can be computed from theelectrostatic potential as

E = −∇V.

2. Electric Field Line

(a) An electric field line is an imaginary curvethat follows the electric field.

(b) If we follow the electric field line, theelectrostatic potential decreases.

(c) The density of electric field lines isproportional to the strength of the electricfield.

3. Equipotential Surface

(a) An equipotential surface is a set of pointswith the same electrostatic potential.

(b) The electric field line is normal to theequipotential surface.

4. Electric field and Conductor

(a) Inside a conductor the electric field vanishes.

(b) Just outside a conductor the electric field isnormal to the conducting surface.

5. Computing Derivatives with DiscreteAlgebraThe derivative of a differentiable function f(x) isdefined by

df(x)

dx= lim

∆→0

f(x + ∆)− f(x)

∆.

(a) The derivative f ′(x) exists if the limits existin both (forward and backward) directionsand they are equal.

f ′+(x) ≡ lim∆→0+

∆+f(x)

∆,

f ′−(x) ≡ lim∆→0−

∆−f(x)

∆.

Here, ∆+f(x) [∆−f(x)] is the forward(backward) difference

∆+f(x) ≡ f(x + ∆)− f(x),

∆−f(x) ≡ f(x)− f(x−∆).

(b) If it is impossible to take the limit ∆→ 0,then we need to develop a method to estimatethe derivative with a systematic error control.

(c) By making use of Taylor’s expansion, we findthat ∆+f(x)/∆ has an error of order ∆ orhigher:

∆+f(x)

∆− f ′(x)

=f(x + ∆)− f(x)

∆− f ′(x)

=1

2f ′′(x)∆ +

∞∑k=3

f (k)(x)∆k−1

k!,

where f (k)(x) is the kth-order derivative off(x).

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(d) The backward difference also has an error oforder ∆ or higher.

(e) The central difference is defined by

∆0f(x) ≡ f(x + 12∆)− f(x− 1

2∆).

The derivative computed by making use of

f ′0(x) ≡ lim∆→0+

∆0f(x)

∆,

has considerably smaller error that is oneorder higher in ∆.

Instrumentation

1. Conductive Paper After measuring the potentialof each point on the paper, we constructequipotential lines and electric field lines in thefollowing cases:

• E1: two points

• E2: one point and one bar

• E3: two bars with a conductive ring

2. Connection

Experimental Procedure

1. Measure the potential of each point of theconductive paper E1, E2, and E3.

2. Open the Excel file.

3. Fill in the Excel file with data.



Note that the colored boxes represent theelectrodes.

4. Click The equipotential line / Draw ,

The electric field line / Draw , and

The electric field / Draw on the proper Excel

sheet.



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Discussion (7 points)

Problem 1Problem 2



Problem 3
