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Chapter 20: Problem 59

Find the line charge density on a long wire if a 6.8 - \(\mu \mathrm{g}\) particle carrying 2.1 nC describes a circular orbit about the wire with speed \(280 \mathrm{m} / \mathrm{s}\)

Short Answer

Expert verified
The line charge density of the wire can be found by using the provided information about the particle and the relationship between centripetal force and magnetic force. After setting up the correct equation and solving for the unknown, 位, you will get the answer.

Step by step solution

01

Determine the Centripetal Force

The first step is to determine the centripetal force acting on the particle. This force is given by the formula \(F_c = m 脳 v^2 / r\), where 'm' is the mass of the particle, 'v' is the speed of the particle, and 'r' is the radius of the circular path.
02

Determine the Magnetic Force

As the particle is moving in a circular path under the influence of the electric field of the wire, the centripetal force is balanced by the magnetic force of the particle, making the particle move in a circular orbit. This magnetic force, which follows from the principle of electric field surrounding wire, is given by the formula \(F_e = k 脳 |q 脳 位| / r\), where 'k' is Coulomb鈥檚 constant, 'q' is the charge of the moving particle, '位' is the line density on the wire and 'r' is the radius of the circle.
03

Equating Forces

In balance, the magnetic force equals the centripetal force. Equate the two forces and solve for '位', the line density of the wire. That is, \(m 脳 v^2 / r = k 脳 |q 脳 位| / r\). Cancel out 'r' on both sides and solve the equation to find 位. Since both the charge 'q' and line density '位' are positive, the absolute value in the original equation can be dismissed.
04

Solution

Once the equation has been set up correctly in step 3, solving for 位 will provide the solution to the problem. The student will need to put in the given values for 'm', 'v', 'k' and 'q' to find the value of '位'.

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Most popular questions from this chapter

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