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Chapter 20: Problem 46

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

Short Answer

Expert verified
The electric field is zero at a point 0.3 m away from the \(1.0-\mu C\) charge, on its side away from the \(2.0-\mu C\) charge. There does not exist a point between the two charges where the electric field is zero.

Step by step solution

01

Notation and Known variables

Let's denote the \(1.0-\mu C\) charge as Q1 and the \(2.0-\mu C\) charge as Q2. The distance between Q1 and Q2 is 10 cm or 0.1 m.
02

Formulate equations based on Coulomb’s Law

Coulomb’s Law governs the electric field E produced by a point charge, related as E = k * Q/ r^2, where k is Coulomb's constant, Q is the charge and r is the distance from the point to the charge. At the point of zero electric field, the magnitudes of the electric fields produced by Q1 and Q2 are equal. Therefore, for a point between the charges (we'll call this Point A), |E1|= |E2|, we get the equation k* Q1/ x^2 = k * Q2/ (0.1 - x)^2 . For a point outside the charges, on the side of Q1 (we'll call this Point B), we get another equation following the same principle, k * Q1/ x^2 = k * Q2/ (0.1 + x)^2 . It's important to note that k cancels out in both equations.
03

Solve equations to get the values of x

Solving these quadratic equations results in two possible values of x which basically represent the distances from Q1. For point A, upon solving, we get a negative value which cannot represent a valid distance in this physical context, and therefore rejected. Thus, there is no point in between the charges where the electric field is zero. For point B, solving the equation yields x as 0.3 m. Therefore, the electric field is zero at a point 0.3 m away from the \(1.0-\mu C\) charge, on the side away from the \(2.0-\mu C\) charge.

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Most popular questions from this chapter

An electron at Earth's surface experiences a gravitational force \(m_{c} g .\) How far away can a proton be and still produce the same force on the electron? (Your answer should show why gravity is unimportant on the molecular scale!)

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