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Q. 43

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Physics for Scientists and Engineers: A Strategic Approach with Modern Physics
Found in: Page 655

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Illustration

Short Answer

FIGURE shows a thin rod of length with total charge .

a. Find an expression for the electric field strength at point on the axis of the rod at distance from the center.

b. Verify that your expression has the expected behavior if .

c. Evaluate at if and .

Part

The expression for the electrical field,

Part

The expression reduces to the expression for the point charge,

Part

The field strength is

See the step by step solution

Step by Step Solution

Step1: Find the expression (part a)

As illustrated in the illustration, move the axis all along rod with origin at the end. The linear charge density is calculated as follows:

It has a charging aspect to it.

Because the wire section is so small, it can be viewed as a point charge. As a result, the field element produced just at point away from the wire along the axis is given by

The equation is integrated as,

Remove everything in integral that is constant.

Step2: Explained by diagram (part a)

Denote,

Introduce a substitution,

The new integration bounds are

Step3: The expression reduces to the expression for the point charge (part b)

In this case, then can be neglected by . so,

This turns out to be an expression for the expected charge far away from the point; the entire wire appears to be a point charge.

Step4: Field strength (part c)

The vector sum of all forces exerted by a field at a particular position within the field on a unit mass, unit charges, unit magnetic pole, and so on from the figure as

(constant)

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