Problem 54 - Master Resource Book in JEE Main Physics

Question

A positively charged ball hangs from a silk thread. We put a positive test charge \(q_{0}\) at a point and measure \(\frac{F}{q_{0}}\) then it can be predicted that the electric field strength \(E\) is (a) \(>F / q_{\mathrm{o}}\) (b) \(=\frac{F}{q_{0}}\) (c) \(<\frac{F}{q_{0}}\) (d) Cannot be estimated

Step-by-Step Solution

Answer

The electric field strength \(E\) is \(=\frac{F}{q_0}\).

Step 1: Understanding the Problem

We have a setup where a positively charged ball is suspended, and we want to measure the electric field using a positive test charge \(q_0\). We are given a value \(\frac{F}{q_0}\) and need to determine the actual electric field strength \(E\).

Step 2: Identifying the Formula for Electric Field

The electric field \(E\) at a point due to a charge is defined by the formula \(E = \frac{F}{q_0}\), where \(F\) is the force experienced by the test charge \(q_0\).

Step 3: Analyzing the Relationship

Since the electric field strength \(E\) is defined as \(\frac{F}{q_0}\), by definition of the electric field in electrostatics, \(E\) is equal to this value. The measurement \(\frac{F}{q_0}\) directly gives the strength of the electric field caused by the charged ball.

Step 4: Conclusion

Since the definition of the electric field is \(E = \frac{F}{q_0}\), the measured electric field strength is exactly this value. Therefore, the answer is (b) \(=\frac{F}{q_0}\).

Key Concepts

Electrostatics

Electrostatics is a branch of physics that focuses on the study of electric charges at rest. It's an essential cornerstone of understanding electricity and magnetism.
Electrons and protons make up electricity, and they have something known as charge. In electrostatics, we study the forces, fields, and potentials associated with such stationary charges.

One key concept in electrostatics is the electric field, which represents how a charge exerts force on other charges nearby. The direction of the electric field is typically the direction in which a positive test charge would be pushed. Understanding these concepts allows you to predict how charges interact without them moving from their fixed positions.

Some critical points in electrostatics include:

• Charges interact with each other based on their signs; like charges repel each other, while opposite ones attract.
• The electric field is a vector field around a charged object, showing the direction and magnitude of the force a charge would experience.
• It's crucial in electrostatics to consider how fields overlap when multiple charges are present.

Coulomb's law

Named after Charles-Augustin de Coulomb, Coulomb's Law is a fundamental principle in electrostatics.
It quantitatively describes the force between two point charges. Essentially, it helps us calculate how strong the electric force will be between two charges.

Coulomb's Law formula is given by:\[ F = k \frac{|q_1 q_2|}{r^2} \]where:

• \( F \) = force between the charges,
• \( k \) = Coulomb's constant \( (8.9875 \times 10^9 \text{ N m}^2/\text{C}^2) \),
• \( q_1 \) and \( q_2 \) = magnitudes of the two charges,
• \( r \) = distance between the centers of two charges.

The formula shows that the force is directly proportional to the product of the two charges and inversely proportional to the square of the distance between their centers.
Simplifying, this means the larger the charges or the closer they are, the greater the force exerted between them. Like charges will repel, and unlike charges will attract, providing a balanced system of forces in electrostatic settings.

Force and Charge Relationship

The relationship between force and charge is at the heart of understanding electric fields. When you place a charge in an electric field, it experiences a force denoted by \( F \).
This fundamental relationship governs much of electrostatics.

Given a charged ball, when you place another test charge \( q_0 \) nearby, the electric field \( E \) at that point causes a force \( F \) on the test charge. Mathematically, this relationship is defined as:\[ E = \frac{F}{q_0} \]

• This equation implies that the field's strength directly depends on the force experienced by the test charge and is inversely proportional to its magnitude.
• If the electric field is stronger, the force on the test charge increases proportionally.
• By rearranging the formula, we understand that force is the electric field strength times the charge \( (F = E \cdot q_0) \).
• Since the test charge \( q_0 \) is usually positive, the force direction shows if the field is attracting or repelling the charge.

Thus, this relationship allows predictions of interactions between charged particles, helping design various technologies and solutions in physics.