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Chapter 20: Problem 60

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
Following through with the simplification and computation in the final step, the line charge density 位 is found to be approximately \(5.6 脳 10^{-8} C/m\).

Step by step solution

01

Recognize the relevant formulas

The two formulas needed to solve this problem are the electric field from a wire \( E = \frac{位}{2蟺蔚鈧�r}\), and the equivalent force due to that field \( F_c = \frac{m v^2}{r} = F = qE\). The constants are the permittivity of free space (\(蔚鈧� = 8.85 脳 10^-12 C^2/N路m^2\)), and the conversion from micrograms to kilograms (\(1 \ 渭g = 10^{-9} \ kg\))
02

Plug values into the centripetal force formula to find the force on the particle

Let's plug the given values into the centripetal force formula. We use the conversion factor to get the mass in kilograms, so the mass \( m = 6.8 渭g = 6.8 脳 10^-9 kg\), the charge \( q = 2.1 \ nC = 2.1 脳 10^{-9} C\) and the speed \( v = 280 \ m/s\). So, \(F_c = \frac{ (6.8 脳 10^{-9} kg) * (280 m/s)^2 } {r}\)
03

Plug values into the formula for force due to an electric field

The force on the particle is also equal to the charge times the electric field, so we have \( F = qE = (2.1 脳 10^{-9} C)E\) where E is the electric field due to the line charge on the wire.
04

Equate the two expressions for force and solve for 位, the line charge density

Next, since both expressions represent the force on the particle, we can equate them and solve for the line charge density 位. To do this, we set the two expressions for force equal to each other: \(\frac{ (6.8 脳 10^{-9} kg) * (280 m/s)^2 } {r} = (2.1 脳 10^{-9} C)E\). Substituting the expression for E, we get \(\frac{ (6.8 脳 10^{-9} kg) * (280 \ m/s)^2 }{r} = 2.1 脳 10^{-9} C * \frac{位}{2蟺蔚鈧�r}\). Therefore, simplifying this we find that 位 = \( \frac{(6.8 脳 10^{-9} kg) * (280 m/s)^2 * 2蟺蔚鈧�}{ (2.1 脳 10^{-9} C)} \)

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