Problem 12 - Master Resource Book in JEE Main Physics

Question

If the electric flux entering and leaving an enclosed surface are \(\phi_{1}\) and \(\phi_{2}\) respectively, then charge enclosed in closed surface is (a) \(\frac{\phi_{2}-\phi_{1}}{\varepsilon_{0}}\) (b) \(\frac{\phi_{1}+\phi_{2}}{\varepsilon_{0}}\) (c) \(\frac{\phi_{1}-\phi_{2}}{\varepsilon_{0}}\) (d) \(\varepsilon_{0}\left(\phi_{2}-\phi_{1}\right)\)

Step-by-Step Solution

Answer

The charge enclosed is \( \varepsilon_{0}(\phi_{2}-\phi_{1}) \), option (d).

Step 1: Understanding Electric Flux and Gauss's Law

Electric flux is the measure of the flow of the electric field through a given surface, and it's represented by the symbol \( \Phi \). According to Gauss's Law, the net electric flux \( \Phi_{net} \) through a closed surface is directly proportional to the enclosed charge \( Q \) and is given by the equation \( Q = \varepsilon_{0} \Phi_{net} \), where \( \varepsilon_{0} \) is the permittivity of free space.

Step 2: Calculate the Net Electric Flux

The net electric flux \( \Phi_{net} \) through the surface is the difference between the electric flux leaving the surface \( \phi_{2} \) and the electric flux entering the surface \( \phi_{1} \). This is calculated as \( \Phi_{net} = \phi_{2} - \phi_{1} \).

Step 3: Apply Gauss's Law to Find Enclosed Charge

According to Gauss's Law, \( Q = \varepsilon_{0} \Phi_{net} \). Substituting the expression for \( \Phi_{net} \) gives \( Q = \varepsilon_{0} (\phi_{2} - \phi_{1}) \).

Step 4: Identify the Correct Option

Comparing the expression \( Q = \varepsilon_{0} (\phi_{2} - \phi_{1}) \) to the options given, the correct answer matches option (d).

Key Concepts

Electric Flux

Electric flux is an essential concept in understanding electric fields and their interactions with surfaces. It measures how much the electric field "flows" through a surface. Picture it like the number of electric field lines passing perpendicularly through a given area. This concept is crucial for analyzing complex electrostatic problems.

One of the most critical applications of electric flux is in Gauss's Law, which relates the electric flux through a closed surface to the charge enclosed within that surface. This relation is integral for solving problems involving symmetrical charge distributions, allowing one to deduce field properties without calculating them directly everywhere.

Enclosed Charge

The enclosed charge in an area is the amount of electric charge located within a closed surface. According to Gauss's Law, we calculate it by determining the net electric flux through the closed surface and relating it to the enclosed charge. That's where Gauss's Law comes in handy:

• Calculate the net electric flux (the electric flux leaving the surface minus the electric flux entering the surface).
• Relate it to the enclosed charge using the equation: \[ Q = \varepsilon_{0} \Phi_{net} \]This relationship simplifies calculating the charge within a complex structure by focusing on the surface's surrounding characteristics.
  
  This quantitative relationship is instrumental in determining charge distribution, which aids in defining scenarios like charged spheres or cylinders.

Permittivity of Free Space

The permittivity of free space, often denoted as \( \varepsilon_{0} \), is a fundamental constant used in physics and engineering. It characterizes the ability of the vacuum to permit electric field lines. This constant is crucial when applying Gauss's Law, as it helps quantify the relation between the electric flux through a surface and the enclosed charge.

The permittivity of free space is a constant value approximately equal to \( 8.854 \times 10^{-12} \) \( \text{F/m} \) (farads per meter). It plays a key role in the equations that describe electromagnetic interactions, reminding us of the intrinsic properties of space that allow for electric fields to propagate.