Decay Rate

Step 1: Understanding the Concept of a Spherical Shell
According to Gauss's Law, the electric potential inside a charged conducting spherical shell is constant and equal to the potential on its surface. However, for an insulating spherical shell, the potential follows a different equation.
Step 2: Given Data
- Radius of the charged shell: cm
- Potential on the surface: V
- Distance from the center where potential is to be found: cm
Step 3: Applying the Concept
For a conducting spherical shell, the potential inside the shell remains constant and is equal to the surface potential: Thus, at cm (inside the shell), However, if the shell is non-conducting, then the potential at any point inside is given by: Since inside a conducting shell, the charge is distributed on the outer surface, the electric field inside is zero, leading to a constant potential.
Step 4: Conclusion
Since the given problem is based on a charged conducting spherical shell, the potential inside is constant. Thus, the potential at cm from the center is:

Step 1: Understanding the Transformer Equation
For a transformer, the relationship between voltage and turns is given by: where:
- and are the primary and secondary voltages,
- and are the number of turns in the primary and secondary coils. Similarly, the current relationship is: where and are the primary and secondary currents.
Step 2: Given Data
- , ,
- V, A.
Step 3: Calculating the Output Current
Using the current equation:
Step 4: Conclusion
Since this is a step-up transformer, the voltage increases while the current decreases. The output current is A.

Step 1: Understanding the Unit of Angular Momentum
The joule-second (Js) is the SI unit of angular momentum, which is a measure of rotational motion.
Step 2: Formula for Angular Momentum
Angular momentum ( ) is defined as: where:
- is the moment of inertia ( ),
- is the angular velocity ( ). From the above relation: which is equivalent to Js (joule-second).
Step 3: Physical Meaning
- Angular momentum represents the rotational equivalent of linear momentum.
- It is a conserved quantity, meaning that in the absence of external torque, the total angular momentum of a system remains constant.
Step 4: Conclusion
Since angular momentum has the unit Js, the correct answer is Angular Momentum.

Step 1: Understanding Electron Emission
Electron emission from a metal surface requires an external influence to overcome the work function ( ), which is the minimum energy required to remove an electron.
Step 2: Role of Electric Field in Electron Emission
- The electric field ( ) required for field emission of electrons is determined by: where:
- is the work function of the metal (in joules),
- is the charge of an electron ( C),
- is the separation distance over which the field is applied. - For most metals, experimental values show that a minimum electric field of approximately: is required to achieve electron emission.
Step 3: Conclusion
The minimum electric field required to emit an electron from a metal is approximately Vm .

Step 1: Understanding Frequency Bands
Radio waves are classified into different bands based on their frequency range.
The main categories include:
ELF (Extremely Low Frequency): Below 3 kHz
VLF (Very Low Frequency): 3 kHz - 30 kHz
LF (Low Frequency): 30 kHz - 300 kHz
MF (Medium Frequency): 300 kHz - 3 MHz
HF (High Frequency): 3 MHz - 30 MHz
VHF (Very High Frequency): 30 MHz - 300 MHz
UHF (Ultra High Frequency): 300 MHz - 3 GHz
SHF (Super High Frequency): 3 GHz - 30 GHz
Step 2: Cellular Communication and the UHF Band
- Cellular phone networks operate in the Ultra High Frequency (UHF) band, which ranges from 300 MHz to 3 GHz.
- This frequency range is preferred for mobile communication because:
- It allows for long-range transmission with minimal interference.
- It can penetrate obstacles like buildings and walls effectively.
- It supports high-speed data transfer. Step 3: Conclusion
Since mobile phones primarily use frequencies within 700 MHz - 2600 MHz, they fall within the UHF band.

Step 1: Understanding Ampere-Maxwell Law
The Ampere-Maxwell Law is a modification of Ampere’s Circuital Law, which originally stated: where:
- represents the circulation of the magnetic field around a closed loop,
- is the conduction current enclosed by the loop,
- is the permeability of free space.
Step 2: Maxwell's Addition - Displacement Current
James Clerk Maxwell introduced the concept of displacement current, which accounts for the changing electric field, leading to the corrected equation: where:
- is the permittivity of free space,
- represents the rate of change of electric flux, contributing to displacement current.
Step 3: Importance of Ampere-Maxwell Law
- This equation bridges the gap between electric and magnetic fields.
- It is one of Maxwell's four equations, crucial for electromagnetism.
- Explains how a changing electric field produces a magnetic field, leading to the concept of electromagnetic waves.

Step 1: Understanding Current Density
Current density ( ) is defined as the electric current per unit area of cross-section: where:
- is the current density,
- is the electric current (in amperes),
- is the cross-sectional area (in square meters).
Step 2: SI Unit of Current Density
- The SI unit of electric current ( ) is ampere (A).
- The SI unit of area ( ) is square meter (m ).
- Thus, the SI unit of current density is:
Step 3: Importance of Current Density
- Higher current density means a stronger electric field in a conductor.
- Used in Ohm's Law in differential form: where is conductivity and is the electric field. Thus, the SI unit of current density is .

Step 1: Understanding the Given Unit
The unit given is volt-second per ampere-meter ( ).
This unit corresponds to the magnetic permeability ( ), specifically the permeability of free space ( ).
Step 2: Definition of Magnetic Permeability
Magnetic permeability ( ) is a measure of how easily a material allows magnetic field lines to pass through it.
The permeability of free space ( ) is a universal constant used in electromagnetism.
Step 3: Expression for
The permeability of free space is given by:
Step 4: Importance of
Appears in Ampère’s Circuital Law and Maxwell’s Equations.
Defines the relationship between magnetic field and current in free space. Thus, the physical quantity represented by is the permeability of free space ( ).