Question

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

Short answer

Answer: The value of the limit as x approaches 10 is approximately 0.0435 N/C.

Step-by-step solution

Identify the Limit

We are asked to find the limit as \(x\) approaches \(10\), i.e. \(\lim_{x \rightarrow 10} E(x)\).

Apply the Limit to the Electric Field Equation

Apply the limit to \(E(x)\): \(\lim_{x \rightarrow 10} \frac{4.35}{x\sqrt{x^2 + 0.01}}\).

Substitute x with the Given Value

Substitute \(x\) with \(10\) in the equation: \(\frac{4.35}{10\sqrt{10^2 + 0.01}}\).

Simplify the Equation

Simplify the expression in the square root first: \(\sqrt{10^2 + 0.01} = \sqrt{100.01}\). Next, compute the value of \(\frac{4.35}{10\sqrt{100.01}}\) by plugging in the simplified square root value which is around \(\frac{4.35}{10\cdot 10.0005}\).

Calculate the Limit

Finally, calculate the value of the expression: \(\frac{4.35}{10\cdot 10.0005}\approx 0.0435\) So, \(\lim_{x \rightarrow 10} E(x) \approx 0.0435\) N/C

Key concepts

Calculus

Calculus is a branch of mathematics that deals with the study of change and motion, through the use of derivatives and integrals. It is immensely useful for solving problems in physics, engineering, economics, and other fields. A good grasp of calculus is necessary to understand concepts like the rate of change of quantities, optimization of systems, and in our particular case, calculating the limit of a function.

In the provided exercise, calculus comes into play when we attempt to find the electric field intensity at a particular point by using limits. The limit process allows us to predict the behavior of a function as the input value gets infinitely close to a certain number, which is essential for understanding continuous phenomena in physical sciences.

Limits

Limits are fundamental in calculus and are used to describe the behavior of a function as its argument approaches a particular value. They help us define the concepts of continuity, derivatives, and integrals. In this context, calculating a limit is like peering into the behavior of a function at a very precise point without necessarily reaching it.

In our exercise, we are looking at the limit of the electric field function as the distance from a charge distribution approaches 10 meters. This approach is especially useful for evaluating functions at points where they cannot be simply substituted due to undefined expressions or more complicated behavior, such as near singularities or points of discontinuity. The purpose of evaluating such a limit is to understand the electric field's behavior as we get very close to this specific distance.

Electric Field Intensity

The electric field intensity is a vector quantity that represents the force per unit charge experienced by a stationary test charge in an electric field. In physics, it's critical to conceptualize how the electric field varies with distance from a given charge distribution. The intensity gives us information about the strength and direction of the electric field.

When we apply calculus to find the electric field intensity, it allows us to not just look at specific distances but also to examine the changing behavior of that field. In the exercise, the function for the electric field intensity depends on the distance from a line of charge. By calculating its limit as we approach a distance of 10 meters, we get insight into the approximate field strength at that point, which has practical implications for understanding electric force in this configuration.

Mathematical Simplification

Mathematical simplification refers to the process of making an equation or expression easier to understand and solve. Simplification often includes reducing fractions, rationalizing denominators, factoring expressions, and canceling common factors. In physics and engineering problems, simplification can reveal underlying patterns and relationships that may not be apparent in the original formulation.

In our electric field example, simplification is a vital step. After substituting the limit value into the function, we must simplify to make the calculation manageable. The operation involves breaking down the square root and performing arithmetic to arrive at an approximate value of the limit. Simplifying complex mathematical expressions can lead to significant time savings and minimize errors in calculation, especially in higher levels of math, making it an indispensable skill.