Problem 17 - Master Resource Book in JEE Main Physics

Question

Three charges each of \(+1 \mu \mathrm{C}\) are placed at the corners of an equilateral triangle. If the force between any two charges be \(F\), then the net force on either charge will be (a) \(\sqrt{2} F\) (b) \(F \sqrt{3}\) (c) \(2 F\) (d) \(3 \underline{F}\)

Step-by-Step Solution

Answer

The net force on a charge is \(F\sqrt{3}\), option (b).

Step 1: Understand the Setup

We have three charges, each with a magnitude of \(+1 \mu \mathrm{C}\), placed at the corners of an equilateral triangle. The side of the triangle is \(r\). The problem statement provides that the force between any two charges is \(F\).

Step 2: Calculate Force Between Two Charges

Using Coulomb's Law, the force between any two charges \(q_1\) and \(q_2\) separated by a distance \(r\) is given by:\[F = \frac{k \times |q_1| \times |q_2|}{r^2}\]In this case, \(q_1 = q_2 = +1 \mu \mathrm{C}\). So, the force is already given as \(F\).

Step 3: Determine Forces Acting on One Charge

Consider one charge, say at point \(A\). The two other charges at \(B\) and \(C\) apply forces on \(A\). Each of these forces is equal to \(F\) and acts along the lines \(AB\) and \(AC\) respectively.

Step 4: Calculate Resultant Force on One Charge

The angle between the forces \(F_{AB}\) and \(F_{AC}\) is \(60^\circ\) because of the equilateral triangle setup. The net force \(F_{net}\) acting on charge at \(A\) can be found using vector addition:\[F_{net} = \sqrt{F^2 + F^2 + 2FF\cos60^\circ}\]\[F_{net} = \sqrt{F^2 + F^2 + FF}\]\[F_{net} = \sqrt{2F^2 + F^2}\]\[F_{net} = \sqrt{3F^2}\]\[F_{net} = F\sqrt{3}\]

Step 5: Interpret the Result

The net force on any one of the charges, due to the two other similar charges in an equilateral triangle, is \(F\sqrt{3}\). Therefore, the correct option is \((b) F\sqrt{3}\).

Key Concepts

Coulomb's Law

Coulomb's Law is a fundamental principle in electrostatics that describes how the force between two charged objects behaves. It tells us that the force (\( F \)) between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance (\( r \)) between them. Mathematically, this is expressed as:\[F = \frac{k \times |q_1| \times |q_2|}{r^2}\]where \( k \) is Coulomb's constant (\( 8.9875 \times 10^9 \, ext{N} \, ext{m}^2 \, ext{C}^{-2} \)). In the context of our problem, the charges are identical, each \(+1 \mu ext{C}\), meaning the force is the same for each pair of charges.

• The force is attractive if the charges are of opposite signs.
• The force is repulsive if the charges are of the same sign, as in our problem.

Coulomb's Law helps us calculate the magnitude of these forces, which is crucial for understanding interactions in an equilateral triangle setup.

Equilateral Triangle

An equilateral triangle is a special type of triangle where all three sides are equal in length, and consequently, all three interior angles are equal, each being\(60^\circ\). This geometric symmetry simplifies calculations when determining forces between charges placed at the corners.

• Each side of the triangle is equal.
• Angles between any two sides are \(60^\circ\).

In our exercise, the setup of the equilateral triangle implies that the forces between the charges are symmetrical. This symmetry allows us to use basic trigonometry and vector addition conveniently. The forces acting on any one charge are at \(60^\circ\) angles to each other, which is a key aspect when summing the forces using vector addition.

Vector Addition

Vector addition is essential when calculating the resultant or net force in scenarios where multiple forces act at angles to each other. In this exercise, two equal forces, each with magnitude \(F\), act on a charge at the vertices of the equilateral triangle.When forces act at an angle (such as \(60^\circ\) here), we use vector addition to find the resultant force. The formula for the resultant force (\(F_{net}\)) due to two vectors of equal magnitude (\(F\)) at an angle \(\theta\) is given by:\[F_{net} = \sqrt{F^2 + F^2 + 2FF\cos\theta}\]Substituting \(\theta = 60^\circ\), we find:\[F_{net} = \sqrt{F^2 + F^2 + FF}\]This method shows how vectors combine geometrically, which is different from simple algebraic addition. Recognizing this difference is a common challenge in physics.

Net Force Calculation

Calculating the net force on a charge involves considering both the magnitude and direction of all forces acting on it. In our equilateral triangle scenario, this means summing the effects of two forces due to adjacent charges.After applying vector addition, the formula simplifies, showing that the net force (\(F_{net}\)) on a charge at one vertex is:\[F_{net} = F\sqrt{3}\]This formula arises from substituting the specific values and characteristics of equilateral triangles into the general vector addition equation.

• The calculation shows that while each pairwise force is \(F\), their combined effect is greater due to the angle between them.
• This result (\(F\sqrt{3}\)) gives the net force magnitude, considering both forces and the geometry involved.

Understanding this concept is vital as it applies in various physics and engineering fields where forces interact in complex setups.