Question

State Gauss’s law. Determine the electric field intensity at a point due to an infinitely long uniformly charged straight wire.

Hint:

Remember standard Gauss law results: Line charge: \(E \propto \frac{1}{r}\) Plane sheet: Constant field Point charge: \(E \propto \frac{1}{r^2}\)

Answer 1

Concept Overview: 
Gauss’s law connects the electric flux passing through a closed surface to the total charge enclosed within that surface. \[ \oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0} \] For charge distributions with high symmetry (such as spherical, cylindrical, or planar symmetry), Gauss’s law allows us to directly determine the electric field. 

Step 1: Statement of Gauss’s Law
The total electric flux through any closed surface is equal to \( \frac{1}{\varepsilon_0} \) times the net charge enclosed by that surface. 

Step 2: Symmetry of an Infinite Line Charge
Consider an infinitely long straight wire carrying uniform charge. - The electric field is directed radially outward (or inward). 
- The magnitude of the field is the same at all points located at the same distance \( r \) from the wire. 
- The system exhibits cylindrical symmetry. 

Step 3: Selection of a Gaussian Surface
Choose a cylindrical Gaussian surface: - Radius = \( r \) 
- Length = \( L \) 
- The wire lies along the central axis of the cylinder. 

Step 4: Calculation of Electric Flux
The electric field is: - Perpendicular to the curved surface of the cylinder. 
- Parallel to the flat circular ends. Therefore, flux through the flat ends is zero. Flux only passes through the curved surface. \[ \Phi = E \times (2\pi r L) \] 

Step 5: Charge Enclosed
If the linear charge density is \( \lambda \), then the charge enclosed within length \( L \) is: \[ Q_{\text{enc}} = \lambda L \] 

Step 6: Apply Gauss’s Law
\[ E(2\pi r L) = \frac{\lambda L}{\varepsilon_0} \] Cancel \( L \) from both sides: \[ E(2\pi r) = \frac{\lambda}{\varepsilon_0} \] \[ E = \frac{\lambda}{2\pi \varepsilon_0 r} \] 

Step 7: Characteristics of the Electric Field
- The magnitude varies inversely with distance \( r \) (i.e., proportional to \( \frac{1}{r} \)). 
- Directed radially outward for positive charge. 
- Directed radially inward for negative charge.