Problem 87 - Calculus Early Transcendentals

Question

The magnitude of the electric field at a point \(x\) meters from the midpoint of a \(0.1-\mathrm{m}\) line of charge is given by \(E(x)=\frac{4.35}{x \sqrt{x^{2}+0.01}}(\text { in units of newtons per coulomb }, \mathrm{N} / \mathrm{C}).\) Evaluate \(\lim _{x \rightarrow 10} E(x)\).

Step-by-Step Solution

Answer

Answer: The approximate limit of the electric field function as x approaches 10 is 0.04349 N/C.

Step 1: Identify the function

The electric field function is given by: \(E(x) = \frac{4.35}{x\sqrt{x^{2}+0.01}}\) Now we need to evaluate the limit as x approaches 10:

Step 2: Evaluate the limit

Evaluate the limit as x approaches 10: \(\lim_{x \rightarrow 10} E(x) = \lim_{x \rightarrow 10} \frac{4.35}{x\sqrt{x^{2}+0.01}}\) As x approaches 10, substitute 10 into the function: \(\lim_{x \rightarrow 10} E(x) = \frac{4.35}{10\sqrt{10^{2}+0.01}}\)

Step 3: Simplify the expression

Calculate the expression inside the square root: \(10^{2}+0.01 = 100+0.01 = 100.01\) Now, calculate the square root: \(\sqrt{100.01} \approx 10.0005\)

Step 4: Calculate the final value of the limit

We substitute the calculated values back into the equation: \(\lim_{x \rightarrow 10} E(x) = \frac{4.35}{10(10.0005)}\) Now, divide the numerator by the product of numbers in the denominator: \(\lim_{x \rightarrow 10} E(x) \approx \frac{4.35}{100.005} \approx 0.04349\) Thus, the limit of the electric field function as x approaches 10 is approximately 0.04349 N/C.

Key Concepts

Electric Field

The concept of the electric field is fundamental in understanding how charged objects interact in space. An electric field is essentially a region around a charged object where a force would be experienced by other charged objects. This force is quantified as the electric field strength, measured in newtons per coulomb (N/C). The electric field due to a positive charge points away from the charge, whereas it points towards a negative charge.
Electric fields have both magnitude and direction, making them vector quantities. The field can be represented visually with field lines, which show the direction a positive test charge would move if placed in the field. In calculations, the formula for an electric field may vary based on the charge distribution, like point charge, line of charge, or surface charge.
In problems involving limits, such as the exercise provided, the goal is often to find the strength of the electric field at a specific point when the charge distribution or configuration is defined by a mathematical expression. Understanding these expressions and their behaviors at specific points is crucial to comprehensively interpret the effect of electric fields on charges.

Line of Charge

Lines of charge represent a linear distribution of charge, where charge is spread along a length of a particular object, like a rod or wire. This can contrast with points or surfaces of charge. For lines of charge, instead of dealing with individual charges, we consider the linear charge density, denoted as \( \lambda \), which is the amount of charge per unit length.
The electric field created by a line of charge depends on distance from the line. The formula can be more complex than that for a point charge due to the geometry of the situation. In a perfect infinite line, the electric field can be derived using Gauss's Law or integral calculus to account for the continuous spread of charge.
In the context of the original exercise, the electric field along a line of charge is addressed by a specific function, which creates a more complicated behavior. Evaluating limits of this function at points near or far from the line can reveal useful information regarding how the electric field interacts with physical space and changes with distance from the line.

Square Root Simplification

Square root simplification is a mathematical skill often required in physics to simplify expressions for ease of calculation. When dealing with formulas involving square roots, it helps to break down the expression by calculating the square root separately and then simplifying the total expression.
In the formula \(E(x) = \frac{4.35}{x\sqrt{x^{2}+0.01}}\), the expression \(\sqrt{x^{2}+0.01}\) requires simplification. Careful approximation at certain values, such as when \(x\) approaches a large number like 10, allows for clearer evaluation of limits.
For practical purposes, methods like rationalizing denominators and estimating square roots nearby whole numbers become important. They can help refine calculations without needing overly complex operations. Accurate simplification of such elements is crucial when determining exact outcomes in scientific problems, helping avoid rounding errors that may accumulate in more detailed computations.