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ECE 2205 - Class Test 01 Solutions

Test Overview

• Course Code: ECE 2205
• Time Allowed: 30 Minutes
• Total Marks: 20

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Question 1: Point Charge inside a Conducting Shell (7 Marks)

Problem Statement

A positive point charge $Q$ is at the center of a spherical conducting shell of inner radius $R_i$ and outer radius $R_o$. Illustrate the variation of Electric field intensity, $E$, and Electric potential, $V$, as a function of radial distance $R$.

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1. Theoretical Analysis & Induced Charges

• Center: Point charge $+Q$ located at $R=0$.
• Inner Surface ($R=R_i$): An induced charge of $-Q$ accumulates uniformly on the inner conducting surface to shield the field inside the conductor.
• Outer Surface ($R=R_o$): An induced charge of $+Q$ accumulates uniformly on the outer conducting surface to maintain neutrality.

Key Electrostatic Principle

Inside any conductor in electrostatic equilibrium, the electric field is strictly zero ($\mathbf{E}=0$), and the conductor forms an equipotential volume ($V=\text{constant}$).

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2. Piecewise Formulations

A. Electric Field Intensity $E(R)$

Using Gauss’s Law (${\oint }_S\mathbf{E}\cdot d\mathbf{S}=\frac{Q_{\text{enc}}}{{\epsilon }_0}$):

• Region I ($0<R<R_i$): $E(R)=\frac{Q}{4\pi {\epsilon }_0{R}^2}$
• Region II ($R_i\le R\le R_o$) [Inside Conductor]: $E(R)=0$
• Region III ($R>R_o$): Enclosed charge $Q_{\text{enc}}=+Q-Q+Q=+Q$: $E(R)=\frac{Q}{4\pi {\epsilon }_0{R}^2}$

B. Electric Potential $V(R)$

Taking the reference potential at infinity ($V(\infty )=0$) using $V(R)=-{\int }_{\infty }^R\mathbf{E}\cdot d\mathbf{R}$:

• Region III ($R\ge R_o$): $V(R)=\frac{Q}{4\pi {\epsilon }_{0}R}$
• Region II ($R_i\le R\le R_o$): $V(R)=V(R_o)=\frac{Q}{4\pi {\epsilon }_0{R}_o}(\text{Constant})$
• Region I ($0<R<R_i$): $V(R)=\frac{Q}{4\pi {\epsilon }_0}\left(\frac{1}{R}-\frac{1}{R_i}+\frac{1}{R_o}\right)$

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3. Graphical Variations

Visualizing $E(R)$ and $V(R)$

               Electric Field Intensity E(R)
   E(R) ^
        |   \
        |    \ 
        |     \
        |      |
        +------+-------+------------------------> R
        0     R_i     R_o
        |<---E=1/R²--->|E=0|<----E=1/R²-------->
 
 
               Electric Potential V(R)
   V(R) ^
        |   \
        |    \ 
        |     \_________
        |      |        \
        |      |         \
        +------+---------+----------------------> R
        0     R_i       R_o
        |<--V ~ 1/R--->|Constant|<---V ~ 1/R---->

Question 2: Fundamental Definitions (6 Marks)

a. Field

A field is a spatial distribution of a physical quantity such that every point in a region of space (and time) can be represented by a mathematical function.

• Scalar Field: Represented by a single magnitude at each point (e.g., Electric Potential $V$, Temperature $T$).

• Vector Field: Represented by both magnitude and direction at each point (e.g., Electric Field Intensity $\mathbf{E}$, Magnetic Flux Density $\mathbf{B}$).

b. Gaussian Surface

A Gaussian surface is an arbitrary, closed 3D mathematical surface chosen in a region of space to evaluate electric flux and apply Gauss’s Law:

${\Phi }_E={\oint }_S\mathbf{E}\cdot d\mathbf{S}=\frac{Q_{\text{enc}}}{{\epsilon }_0}$

Design Criteria

To simplify integration, a Gaussian surface is chosen to reflect the underlying charge symmetry (e.g., spherical for point charges, cylindrical for line charges, planar for infinite sheets).

c. Equipotential Line

An equipotential line (or surface in 3D) is a spatial locus of points all possessing the exact same electric potential ($V=\text{constant}$).

Core Properties

1. Zero Work: Moving a test charge along an equipotential line requires zero net work ($W=q\Delta V=0$).

2. Orthogonality: Electric field lines are always perpendicular ($\perp$) to equipotential lines at every point of intersection ($\mathbf{E}=-\nabla V$).

Question 3: Fundamental Postulates of Electrostatics (4 Marks)

Integral Forms in Free Space

1. Gauss’s Law for Electrostatics (First Postulate)

${\oint }_S\mathbf{E}\cdot d\mathbf{S}=\frac{Q}{{\epsilon }_0}=\frac{1}{{\epsilon }_0}{\int }_V{\rho }_{v}dV$

• Meaning in Words: The net outward electric flux through any closed surface $S$ in free space is proportional to the total electric charge enclosed within the volume $V$ bounded by $S$, divided by the permittivity of free space (${\epsilon }_0$).

2. Conservative Property of Electrostatic Field (Second Postulate)

${\oint }_L\mathbf{E}\cdot d\mathbf{l}=0$

• Meaning in Words: The circulation (or line integral) of the electrostatic field intensity $\mathbf{E}$ around any closed path $L$ in free space is identically zero. This implies that electrostatic fields are conservative and irrotational ($\nabla \times \mathbf{E}=0$).

Question 4: Field & Equipotential Lines Sketch (3 Marks)

Problem Statement

Make a two-dimensional sketch of the electric field lines and the equipotential lines of a uniform charge sphere with volume charge density of $-{\rho }_v$.

Key Properties for the Diagram

1. Negative Charge (${\rho }_v<0$): Electric field lines point radially inward toward the center.

2. Equipotential Lines: Form concentric circles centered around the origin.

3. Orthogonality: Field lines and equipotential lines intersect at $90^{\circ }$.

2D Schematic

Plaintext

                        \       |       /
                         \      |      /
                          v     v     v
                        .---.-------.---.
                       /    /   |   \    \
                      /    /    v    \    \
                     |    |   ( O )   |    |  <--- Concentric Dashed Circles
                      \    \    ^    /    /        = Equipotential Lines
                       \    \   |   /    /
                        '---'---+---'---'
                          ^     ^     ^
                         /      |      \
                        /       |       \

         Legend:
         ───> Solid arrows   : Electric Field Lines (Radially Inward)
         ---- Dashed circles  : Equipotential Lines (Concentric)