Question

Two electric charges \(12 \mu \mathrm{c}\) and \(-6 \mu \mathrm{c}\) are placed \(20 \mathrm{~cm}\) apart in air. There will be a point \(P\) on the line joining these charges and outside the region between them, at which the electric potential is zero. The distance of \(P\) from \(-6 \mu c\) charge is ..... (A) \(0.20 \mathrm{~m}\) (B) \(0.10 \mathrm{~m}\) (C) \(0.25 \mathrm{~m}\) (D) \(0.15 \mathrm{~m}\)

Short answer

The distance of point \(P\) from the \(-6 \mu C\) charge is 0.20 m. The correct answer is (A).

Step-by-step solution

Identify Known and Unknown Variables

The known variables from the problem are the electric charges \(Q_1 = 12 \mu C\) and \(Q_2 = -6 \mu C\) and the distance between them \(d = 20 cm\). The point \(P\) lies outside the region between them, where the electric potential is zero. The unknown variable is the distance \(r\) from charge \(Q_2\) to point \(P\).

Write Down the Formula

The formula for the electric potential at a point due to a point charge is \(V = \frac{kQ}{r}\). In this case, we need to consider the electric potentials from both charges and set the total potential to zero. This gives us the equation \(\frac{kQ_1}{r_1} + \frac{kQ_2}{r_2} = 0\).

Substitute the Known Variables into the Equation

Substituting the known variables gives \(\frac{kQ_1}{r_1} + \frac{kQ_2}{r_2} = 0\). Substituting \(Q_1 = 12 \mu C\), \(Q_2 = -6 \mu C\), and \(r_2 = r\) (the unknown distance from \(Q_2\) to point \(P\)) into the equation gives: \[ \frac{k \cdot 12}{d + r} - \frac{k \cdot 6}{r} = 0 \] Note that \(d = 20 cm = 0.2 m\) is the distance between the two charges, so \(r_1 = d + r = 0.2m + r\). The \(k\) terms look like they will cancel out in the next step.

Simplify the Equation and Solve for the Unknown

Dividing the whole equation by \(k\) and then multiplying by \(r(d + r)\) to eliminate the denominators gives: \[ 12r - 6(d + r) = 0 \] Solving this equation for \(r\) gives: \[ r = \frac{6d}{6} = 0.2m \] Therefore, the distance of point \(P\) from the \(Q_2\) charge is 0.20 m. The correct answer is (A).

Key concepts

Point Charges

Point charges are specific types of electric charges that are assumed to occupy a single point in space. This simplification allows us to calculate electric potentials and forces using straightforward formulas. Point charges can have either a positive or negative charge.

Positive point charges, like our 12 µC charge, tend to repel other positive charges and attract negative charges. Negative point charges, like the -6 µC charge, have the opposite effect. These characteristics are critical when determining interactions and arranging charges in space.

• Point charges in problems like this are typically used to simplify calculations.
• They assume charges are very small or concentrated at a point, so their dimensions do not impact calculations significantly.

Understanding how point charges interact is fundamental for solving related problems and predicting potential behaviors in electrical systems.

Electric Charge Interaction

Electric charge interaction refers to how charges affect each other through their electric fields. The principle is simple: like charges repel, unlike charges attract. This interaction is fundamental to understanding electric forces and potentials.

In this exercise, the two charges are opposite, creating a region where their electric potentials can cancel each other out. This is due to the additive nature of potentials; positive potential from the 12 µC charge can be offset by the negative potential from the -6 µC charge when properly positioned along the line joining them.

• Electric charges create fields that exert forces on other charges located within those fields.
• The superposition principle allows us to calculate the net electric potential from multiple charges by adding their individual potentials.

Recognizing how these interactions occur helps in solving where the electric potential will be zero, as it involves balancing these effects precisely.

Coulomb's Law

Coulomb's Law is a fundamental principle describing the force between two point charges. It states that the electrostatic force \(F\) between two charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them:\[ F = k \frac{|Q_1 Q_2|}{r^2} \] Where \(k\) is Coulomb's constant. Although the equation emphasized in solving the original exercise directly involves potentials, Coulomb’s Law is the foundation.

• The principle behind using potentials is derived from integrating Coulomb’s Law, resulting in the electric potentials and their use to find zero points.
• Using Coulomb’s Law helps us understand why, as the distance increases, the force and potential from a charge diminish.

The original problem involves balancing the influence of two charges to find a point where their potentials cancel out (zero net potential)—a direct application of concepts built on Coulomb's Law.