The question asks to identify the law that governs the force between two point charges. This is a fundamental concept in electrostatics. We need to recall the basic laws of electromagnetism and their applications.
Correct Option: D) Coulomb's Law is the fundamental law that quantifies the electrostatic force between two point charges.
The question asks for the approximate value of Coulomb's constant (k) in SI units. This is a fundamental constant in electrostatics, and its value is derived from the permittivity of free space.
C) \(9 \times 10^9 \text{ N} \cdot \text{m}^2/\text{C}^2\) is the standard approximate value for Coulomb's constant in SI units.
This problem involves Coulomb's Law, which describes the electrostatic force between two point charges. We need to analyze how this force changes when the distance between the charges is altered.
Correct Option: D) One-fourth
This problem involves Coulomb's Law, which describes the electrostatic force between two charged particles. We need to analyze how changes in charge magnitudes and the distance between them affect the force.
Recall Coulomb's Law: The electrostatic force (\(F\)) between two point charges (\(q_1\) and \(q_2\)) separated by a distance (\(r\)) is given by:
\[ F = k \frac{q_1 q_2}{r^2} \]where \(k\) is Coulomb's constant.
Define original conditions: Let the original charges be \(q_1\) and \(q_2\), and the original distance be \(r\). The original force (\(F_{\text{original}}\)) is:
\[ F_{\text{original}} = k \frac{q_1 q_2}{r^2} \]Define new conditions: According to the problem statement:
Calculate the new force (\(F_{\text{new}}\)): Substitute the new conditions into Coulomb's Law:
\[ F_{\text{new}} = k \frac{q_1' q_2'}{(r')^2} \] \[ F_{\text{new}} = k \frac{(2q_1)(2q_2)}{\left(\frac{r}{2}\right)^2} \]Simplify the expression for \(F_{\text{new}}\):
\[ F_{\text{new}} = k \frac{4q_1 q_2}{\frac{r^2}{4}} \] \[ F_{\text{new}} = k \frac{4q_1 q_2}{1} \times \frac{4}{r^2} \] \[ F_{\text{new}} = 16 \left( k \frac{q_1 q_2}{r^2} \right) \]Compare \(F_{\text{new}}\) with \(F_{\text{original}}\): We can see that the term in the parenthesis is \(F_{\text{original}}\).
\[ F_{\text{new}} = 16 F_{\text{original}} \]Thus, the new force becomes 16 times the original force.
Correct Option: C) 16 times
The question asks for the SI unit of electric charge. We need to recall the standard SI units for fundamental electrical quantities and forces.
Correct Option: D) Coulomb is the SI unit of electric charge. One coulomb is defined as the amount of charge transported by a constant current of one ampere in one second. Mathematically, this can be expressed as:
\[ Q = I \times t \]
Where \(Q\) is charge, \(I\) is current, and \(t\) is time. So, \(1 \text{ Coulomb} = 1 \text{ Ampere} \times 1 \text{ second}\).
The question asks for the value of the permittivity of free space, denoted as \(\varepsilon_0\). This is a fundamental physical constant used in electromagnetism, particularly in Coulomb's Law and Maxwell's equations. We need to recall its standard value and units.
D) 8.854 × 10⁻¹² C²/N·m² is the internationally accepted value for the permittivity of free space \(\varepsilon_0\).