Electric Charge

Here , and are positive
Therefore V > 0
So , the correct option is (A) : Net potential of the system cannot be zero at a point but net electric field can be zero at that point.

To determine the velocity of the electron, we follow these steps:
1. Electric Field due to an Infinite Cylindrical Wire:
The electric field at a distance from an infinite wire with linear charge density is given by: where is the permittivity of free space ( C /N·m ).
2. Force on the Electron:
The electrostatic force acting on the electron is: where is the charge of the electron ( C).
3. Centripetal Force:
For the electron to move in a circular path, the electrostatic force must provide the necessary centripetal force: Equating the two expressions for : Simplifying, we get:
4. Substitute the Given Values:
Plugging in the values:
Final Answer The velocity of the electron is m/s.

The drift velocity ( ) is given by: where:
,
(number density),
(charge of an electron),
(cross-sectional area).
Rearrange to solve for :
Substitute the values:
Simplify:
Final Answer: The drift velocity is:

Let's analyze the circuit step-by-step to calculate the equivalent resistance between points A and B.

1. Simplify the top two resistors: The two 8-ohm resistors are in series. Their combined resistance is 8 Ω + 8 Ω = 16 Ω.
2. Simplify the two 4-ohm angled resistors: The two 4-ohm resistors connected to the terminals of 8 ohm resistors are also in series. The combined resistance is 4 Ω + 4 Ω = 8 Ω.
3. Simplify the three 4-ohm and combined resistors: The combined 16-ohm resistor, the combined 8-ohm resistor and the 4-ohm resistor are connected in parallel. The formula for parallel resistors is 1/R_eq = 1/R₁ + 1/R₂ + 1/R₃ , so, 1/R_eq = 1/16 + 1/8 + 1/4= (1+2+4)/16 = 7/16 The equivalent resistance is R_eq =16/7

Given that the equivalent resistance is ^x/₇ ohms and calculated equivalent resistance is ¹⁶/₇, so the value of x is 16.

1. Force Balance Condition: - The electrostatic force on the proton is zero when:
   • Using Coulomb’s law:
2. Simplify: - Cancel and :
   • Take the square root:
   • Cross-multiply:
   • Solve for :
3. Position of Proton: - The proton is from the origin.

Final Answer: 24 cm

Step 1: Use Gauss's Law.

According to Gauss's law: That means the electric flux through any closed surface depends only on the net charge enclosed by that surface.

Step 2: Identify which charges are enclosed within the sphere.

The sphere is centered at the origin with radius , so its surface extends from:

Now check the given charges:

• Charge is at : outside the sphere (since ).
• Charge is at : inside the sphere (since ).

Hence, the net enclosed charge inside the sphere:

Step 3: Apply Gauss’s law.

So, comparing with given expression: We get:

Final Answer: