Problem 56 - Calculus Volume 3

Question

The strength \(E\) of an electric field at point \((x, y, z)\) resulting from an infinitely long charged wire lying along the \(y\) -axis is given by \(E(x, y, z)=k / \sqrt{x^{2}+y^{2}},\) where \(k\) is a positive constant. For simplicity, let \(k=1\) and find the equations of the level surfaces for \(E=10\) and \(E=100\).

Step-by-Step Solution

Answer

Level surfaces equations are \(x^2 + z^2 = \frac{1}{100}\) for \(E=10\) and \(x^2 + z^2 = \frac{1}{10000}\) for \(E=100\).

Step 1: Understand Level Surfaces

Level surfaces are surfaces where the function, in this case the strength of the electric field \(E(x, y, z)\), takes on a constant value. Here, we need to find the surfaces where \(E = 10\) and \(E = 100\).

Step 2: Substitute Variables in Electric Field Equation

Given the function \(E(x, y, z) = \frac{1}{\sqrt{x^2 + z^2}}\) (since the wire lies along the \(y\)-axis), substitute \(k = 1\) and set the expression equal to 10 for \(E = 10\): \(\frac{1}{\sqrt{x^2 + z^2}} = 10\).

Step 3: Solve for Level Surface Equation when E = 10

Solve the equation \(\frac{1}{\sqrt{x^2 + z^2}} = 10\) by rearranging it to \(\sqrt{x^2 + z^2} = \frac{1}{10}\), thus finding the level surface equation \(x^2 + z^2 = \frac{1}{100}\).

Step 4: Substitute Variables for E = 100

Use the same process for \(E = 100\): Set \(\frac{1}{\sqrt{x^2 + z^2}} = 100\) and rearrange to find the equation of the level surface.

Step 5: Solve for Level Surface Equation when E = 100

Solve the equation \(\frac{1}{\sqrt{x^2 + z^2}} = 100\) by rearranging it to \(\sqrt{x^2 + z^2} = \frac{1}{100}\), thus finding the level surface equation \(x^2 + z^2 = \frac{1}{10000}\).

Key Concepts

Level Surfaces

Level surfaces are important in understanding how a function behaves in space. Imagine you have a function that depends on three dimensions:

• The level surface represents all points in space where this function takes the same value.
• For example, consider the surface of water in a perfectly still swimming pool. It is at the same height everywhere, representing a level surface.
• In the context of electric fields, a level surface means any point on this surface has the same electric field strength.

With level surfaces, we can visualize how an electric field varies in space. Knowing this helps predict the behavior of charged particles, and in our exercise, it means finding where the electric field strength is constant.

Charged Wire

A charged wire, especially when imagined to be infinitely long, creates a distinctive pattern of electric fields around it. Here are its key features:

• The wire itself, charged with electricity, generates electric fields emanating outward in a circular pattern.
• For simplification, if the wire is aligned along the y-axis, the fields will spread out in the xz-plane, displaying symmetry about the wire.
• The electric field intensity decreases with the square of the distance from the wire, being stronger closer and weaker farther away.

This setup allows us to apply the concept of level surfaces and visualize where the electric field magnitude remains constant as one moves around this wire.

Equations of Surfaces

Equations of surfaces help us to mathematically describe the shape and position of these level surfaces in three-dimensional space. Let's explore this further:

• The equation derived from the electric field formula helps identify surfaces where the electric field magnitude is constant.
• For instance, if the function for electric field strength is given by \( E(x, y, z) = \frac{1}{\sqrt{x^2 + z^2}} \), setting \( E = 10 \) gives us the equation \( x^2 + z^2 = \frac{1}{100} \).
• This equation represents a flat circular surface in the xz-plane at equal distance from the wire, indicating a constant electric field magnitude.

The simplicity of these equations allows for easy calculations and predictions concerning the electric field's influence in various scenarios.

Electric Field Strength

Electric field strength, denoted as \( E \), provides an understanding of how strong the electric forces are at a given point. Here are the essentials:

• It's a measure of the force that a unit positive charge would experience at a particular location.
• The strength depends on factors like distance from the source charge and the configuration of these charges.
• Especially in the context of a linear charge like our wire, the electric field decreases with increased distance from the source, following an inverse relationship.

For practical calculations, substituting values into the field equation allows us to find geographic points (or entire surfaces) where the electric field maintains specific, equal magnitudes. This forms a base for numerous practical applications in sciences and engineering.