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?

When studying electric fields in physics, you may come across the concept of an 'infinite line charge.' Imagine a wire that stretches out infinitely in both directions, uniformly coated with electrical charge. Although truly infinite objects don't exist in the real world, this hypothetical construct simplifies calculations and helps us understand electric fields generated by very long, thin charged objects, akin to our strand of DNA in the exercise.

For an infinite line charge, the electric field is directed radially outward (or inward in the case of negative charge) from the line. The strength of this field diminishes as you move away from the line, specifically inversely proportional to the radial distance. This is different from the electric field created by a point charge, which spreads out spherically and diminishes more quickly—proportional to the square of the distance from the charge according to Coulomb's law.

The electric charge per unit length, often represented by the symbol \( \lambda \), is a measure of how much electric charge is distributed over a linear material, like a wire or, in our exercise, a strand of DNA. The DNA carries a positive charge, where \( +e \) represents the elementary charge per nanometer of the DNA's length.

Understanding this concept is crucial in calculating the electric field produced by the DNA. In our exercise, we treat the strand as if it were infinitely long, which lets us use a simplified equation for the electric field strength around a line charge. To carry out the computation accurately, it's important to ensure that units are consistent, converting our given values to a common unit, such as nanometers, before inserting them into the electric field formula.

Coulomb's Law is a fundamental principle in electromagnetism that describes the force between two point charges. Stated by Charles-Augustin de Coulomb in the 18th century, it tells us that the electric force between two static charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.

The law is encapsulated by the equation \( F = k \frac{q_1 \cdot q_2}{r^2} \), where \(F\) is the force between charges, \(q_1\) and \(q_2\) are the amounts of the charges, \(r\) is the distance between the centers of the two charges, and \(k\) is Coulomb's constant. Although our textbook exercise deals with an infinite line charge rather than point charges, Coulomb's law's principles underpin our understanding of the electric force and fields and are invaluable for solving a wide range of electrostatic problems.