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Chapter 20: Problem 62

A 5.0 - \(\mu\) m strand of DNA carries charge \(+e\) per nm of length. Treating it as a charged line, what's the electric field strength \(25 \mathrm{nm}\) from the DNA, not near either end?

Short Answer

Expert verified
The strength of the electric field \(25 nm\) from the DNA strand is \(5.74 * 10^6 N/C\).

Step by step solution

01

Identify the given parameters

We are given the charge density, which is \(+e/\mathrm{nm}\) (where \(e\) is the charge of an electron, \(1.6*10^{-19} C\)). This is a line charge, so we need to consider the appropriate part of Coulomb's law. It is also given that the point of interest is \(25 nm\) away from the DNA.
02

Align the charge and distance units

Our distance is in nm and our charge is in charges per nm, yet Coulomb's law usually measures in meters and Coulombs. To address this, we can convert our measurements to meters by remembering that \(1 nm = 10^{-9} m\). Thus, the distance from the DNA strand is \(25 * 10^{-9} m\), and our charge density is \(+e * 10^9 C/m\).
03

Apply the formula for the electric field due to a line charge

The formula is \(E = \frac{k\lambda}{r}\), where \(k\) is Coulomb's constant (\(8.99*10^9 N*m^2/C^2\)), \(\lambda\) is the linear charge density (charge/meter), and \(r\) is the distance from the line charge to the field point. Plugging in our given and derived values, we find \(E = \frac {(8.99 *10^9 N*m^2/C^2)(e * 10^9 C/m)}{25 * 10^{-9} m}\).
04

Simplify and solve

Simplifying, we have \(E = \frac {(8.99 *10^9)(10^9)(1.6*10^{-19})}{25 * 10^{-9}} N/C = 5.74 * 10^6 N/C\).

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Most popular questions from this chapter

A proton moving to the right at \(3.8 \times 10^{5} \mathrm{m} / \mathrm{s}\) enters a region where a \(56-\mathrm{kN} / \mathrm{C}\) electric field points to the left. (a) How far will the proton get before it momentarily stops? (b) Describe its subsequent motion.

A free neutron is unstable and soon decays to other particles, one of them a proton. Must there be others? If so, what electric properties must it or they have?

A 1.0 - \(\mu\)C charge and a 2.0 - \(\mu\)C charge are 10 cm apart. Find a point where the electric field is zero.

A molecule has its dipole moment aligned with a 1.2 -kN/C electric field. If it takes \(3.1 \times 10^{-27} \mathrm{J}\) to reverse the molecule's orientation, what's its dipole moment?

A proton is on the \(x\) -axis at \(x=1.6 \mathrm{nm}\). An electron is on the \(y\) -axis at \(y=0.85 \mathrm{nm} .\) Find the net force the two exert on a helium nucleus (charge \(+2 e\) ) at the origin.
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