Problem 61 - University Physics with Modern Physics

Question

(a) Sketch the electric field lines for an infinite line of charge. You may find it helpful to show the field lines in a plane containing the line of charge in one sketch and the field lines in a a plane perpendicular to the line of charge in a second sketch. (b) Explain how your sketches show (i) that the magnitude \(E\) of the electric field depends only on the distance \(r\) from the line of charge and (ii) that \(E\) decreases like \(1 / r .\)

Step-by-Step Solution

Answer

The electric field lines radiate outward and show that the field strength depends on \(r\) and decreases as \(1/r\).

Step 1: Visualize the Geometry

Understand that an infinite line of charge can be imagined as an endless string of charge extending in both directions. This means our line of charge has symmetry along its length, making our analysis focused on radial distance from the line only.

Step 2: Sketch the Electric Field Lines in a Plane Containing the Line

In a plane containing the line of charge, the electric field lines originate radially outward from the line. Because of symmetry, these lines are straight and extend perpendicular to the line of charge indefinitely. Remember, field lines never cross each other.

Step 3: Sketch the Electric Field Lines in a Perpendicular Plane

In a plane perpendicular to the line of charge, you can imagine the field lines radiating out from a series of points along the line. The sketch would appear as radial lines uniformly spaced like the spokes of a wheel all pointing away from where the line charge intersects the plane.

Step 4: Explain Dependence of Magnitude on Distance

Since field lines are straight and the same at all points equidistant from the line of charge, the pattern shows that the magnitude of the electric field depends only on the distance from the line. Greater density of lines (closer to the line) indicates a stronger field, supporting the notion that the field varies with distance.

Step 5: Show Decrease of Electric Field as 1/r

For an infinite line of charge, the density of the electric field lines illustrates that the field strength decreases as you move away from the line. The spacing between lines increases linearly with distance, demonstrating mathematically that the electric field decreases as the inverse of the distance, or as \(1/r\).

Key Concepts

Infinite Line of Charge

Imagine an infinite line of charge as a line that stretches endlessly in both directions. This concept is crucial because it introduces the idea of symmetry. A key feature of an infinite line of charge is that it exhibits continuous uniform charge distribution along its length. This symmetry simplifies our analysis, allowing us to focus solely on the radial aspects of the electric field around it. You won't need to worry about what happens at the "ends" of the line, because there are no ends. By assuming the line of charge is infinite, we can conclude that the electric field has the same characteristics at any point along the line. The nature of the infinite line of charge forms the basis of understanding how electric fields behave in its vicinity, and why the problem narrows to radial considerations alone.

Electric Field Magnitude

The electric field magnitude around an infinite line of charge is essential to understanding how strong the electric force is at any given point in space. To visualize this, imagine the electric field lines radiating outwards from the line of charge. These lines are straight and problematically extend infinitely in radial directions. The magnitude of the electric field at any point depends on how densely packed these field lines are. Closer to the line, the lines are denser, indicating a stronger electric field. This is the clue that tells us the magnitude depends on proximity to the charge. The mathematical relationship can be expressed using the formula: \[ E = \frac{\lambda}{2 \pi \varepsilon_0 r} \] where \( \lambda \) is the linear charge density, \( \varepsilon_0 \) is the permittivity of free space, and \( r \) is the radial distance from the line.

Electric Field and Distance

When it comes to understanding how the electric field interacts with distance, it's crucial to recognize the inverse relationship it holds. As you move further away from the infinite line of charge, the electric field's strength diminishes. This dependency is visualized through the increasing spacing between electric field lines as one moves away from the line. The field's decrease with distance follows an inverse relationship with the radial distance \( r \). In simpler terms, if you double your distance from the line, the electric field strength decreases by half. This follows the expression \[ E \propto \frac{1}{r} \] which signifies that the electric field strength is inversely proportional to the radial distance from the line of charge. Observing this interaction gives us an insightful view of how electric fields behave over varying distances from a linear charge source.

Electric Field Symmetry

Electric field symmetry is a powerful concept that simplifies analyzing complex electric field problems. In the case of an infinite line of charge, symmetry plays a pivotal role. The symmetry here refers to the fact that the electric field is the same at any point at a given radial distance from the line, regardless of the angle around the line. This uniformity arises due to the infinite length and uniform charge distribution. Consequently, when we draw electric field lines in a plane perpendicular to the line, they appear evenly spaced, like spokes on a wheel. The radial nature of the electric field lines showcases this perfect symmetry, which arises due to the consistent charge distribution along the length. This understanding of symmetry not only helps with visualization but also simplifies the mathematical treatment of the electric field near an infinite line of charge.