Chapter 21.1, Problem 7PP

(a)

To determine

Whether the electrostatic force between an unknown charge is same with the two given test charges.

Answer to Problem 7PP

No. The force on the $2.0\times {10}^{-6}\text{ C}$ charge will be twice than that on the $1.0\times {10}^{-6}\text{ C}$ charge.

Explanation of Solution

Introduction:

Electrostatic force between the two charges is directly related to size of both the charges. According to Coulomb, the electrostatic force $\left(F\right)$ between the two objects varies directly with the charge of the two objects and inversely with the square of the distance between the two objects. If $A$ and $B$ are the two objects having charges $q_A$ and $q_B$ respectively which are separated by a distance $r$ , then Coulomb’s law can be written as,

  $F\text{ = K}\frac{q_A{q}_B}{r^2}$

Where $\text{ K}$ is a constant, $\text{ K}=9\times {10}^9\text{ N}\cdot {\text{m}}^{\text{2}}{\text{/C}}^{\text{2}}$

Since the electrostatic force between the two charges is directly related to the product of their charge, the electrostatic force between an unknown charge is not identical with the two given test charges. The force on the $2.0\times {10}^{-6}\text{ C}$ charge will be double than that on the $1.0\times {10}^{-6}\text{ C}$ charge.

Conclusion:

The electrostatic force between an unknown charge is not same with the two given test charges. The force on the $2.0\times {10}^{-6}\text{ C}$ charge will be twice than that on the $1.0\times {10}^{-6}\text{ C}$ charge.

(b)

To determine

Whether the electric field measured between an unknown charge with the two given test charges are equal.

Answer to Problem 7PP

Yes

Explanation of Solution

Introduction:

Electric field is a physical quantity that gives relationship between the force exerted on a test charge and the size of a test charge. It is a vector quantity. If $F_{onq}$ is the electrostatic force on the test charge $q$ at some point, the electric field $\left(E\right)$ on that charge at the same point is given by,

  $E=\frac{F_{onq}}{q}$

Electric force is directly related to the test charge and hence the electric force is doubled if the charge is doubled. But the electric field is the ratio of the electric force to the test charge. Therefore, electric field on a charge is independent of its size. Thus, the electric field measured between an unknown charge with the two test charges $1.0\times {10}^{-6}\text{ C}$ and $2.0\times {10}^{-6}\text{ C}$ are equal.

Conclusion:

The electric field measured between an unknown charge with the two test charges $1.0\times {10}^{-6}\text{ C}$ and $2.0\times {10}^{-6}\text{ C}$ are equal.