Given below are two statements : One is labelled as Assertion \(A\) and the other is labelled as Reason R
Assertion A : Two metallic spheres are charged to the same potential One of them is hollow and another is solid, and both have the same radii Solid sphere will have lower charge than the hollow one.
Reason R : Capacitance of metallic spheres depend on the radii of spheres.
In the light of the above statements, choose the correct answer from the options given below.
Assertion A : Two metallic spheres are charged to the same potential One of them is hollow and another is solid, and both have the same radii Solid sphere will have lower charge than the hollow one.
Reason R : Capacitance of metallic spheres depend on the radii of spheres.
In the light of the above statements, choose the correct answer from the options given below.
Show Hint
For questions involving assertions and reasons, analyze each statement indepen dently. Recall the relevant formulas and concepts to determine the truth or false hood of each statement.
Both \(A\) and \(R\) are true and \(R\)is the correct explanation of \(A\)
Both \(A\) and \(R\) are true but \(R\) is not the correct explanation of \(A\)
A is true but \(R\) is false
A is false but \(R\) is true
The Correct Option is D
Solution and Explanation
Step 1: Analyze Assertion A
The potential of a conducting sphere (solid or hollow) is given by: \[ V = \frac{KQ}{R} \] where \(V\) is the potential, \(K\) is a constant, \(Q\) is the charge, and \(R\) is the radius.
If two spheres have the same potential (\(V_1 = V_2\)) and the same radius (\(R_1 = R_2\)), then: \[ \frac{KQ_1}{R_1} = \frac{KQ_2}{R_2} \] Since \(R_1 = R_2\), it follows that \(Q_1 = Q_2\). Therefore, the assertion that the solid sphere will have a lower charge is false. The charges must be equal.
Step 2: Analyze Reason R
The capacitance of a spherical conductor is given by: \[ C = 4\pi \epsilon_0 R \] where \(C\) is the capacitance, \(\epsilon_0\) is the permittivity of free space, and \(R\) is the radius. This shows that the capacitance of a metallic sphere depends only on its radius. Therefore, the reason is true.
Conclusion: Assertion A is false, but Reason R is true. Therefore, the correct option is (4).
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Concepts Used:
Electric charges and field
What is Electric Charge
It is the property of subatomic particles that experiences a force when put in an electric and magnetic field.
What is Electric Field
It is a property associated with each point in space when charge is present in any form. The magnitude and direction of the electric field are expressed by E, called electric field strength or electric field intensity.
Electric charges are of two types: Positive and Negative. It is commonly carried by charge carriers protons and electrons.
Properties of Electric Charge
Various properties of charge include the following :-
- Additivity of Electric Charge
- Conservation of Electric Charge
- Quantization of Electric Charge
Types of electric charge
Two kinds of electric charges are there :-
Negative Charge - When an object has a negative charge it means that it has more electrons than protons.
Positive Charge - When an object has a positive charge it means that it has more protons than electrons.
When there is an identical number of positive and negative charges, the negative and positive charges would cancel out each other and the object would become neutral.