Electric Charges And Fields

Calculation of Charge (q)

The formula for torque (τ) is given as:

We can also express torque as:

Rearranging to solve for charge (q):

Substituting the given values:

Simplifying the expression:

Conclusion:

The charge (q) is 2 mC.

To solve this problem, we need to understand the behavior of electric charge distribution between two connected spherical conductors.

When two charged spherical conductors are connected by a wire, they come to the same potential because the wire allows charges to move freely between them until equilibrium is reached.

The potential of a charged conductor is given by:

where:

• is Coulomb's constant.
• is the total charge on the conductor.
• is the radius of the spherical conductor.

For both spheres to have the same potential , we have:

Canceling from both sides:

Rearranging gives us the charge relation:

The surface charge density is given by:

Therefore, the surface charge densities for the two spheres are:

Substitute the expression for from the charge relation:

Thus, the ratio of surface charge densities is:

Therefore, the correct answer is:

This confirms that the ratio of surface charge densities is determined by the inverse of the radii ratio of the spheres.

• Conductors: In conductors, such as metals, resistance typically increases with temperature. This is because higher temperatures cause atoms to vibrate more intensely. These vibrations cause increased collisions with the conduction electrons, which hampers their flow, leading to increased resistance. The relationship is often linear and can be represented by the equation: R = R₀(1 + αT), where R is the resistance at temperature T, R₀ is the resistance at a reference temperature, and α is the temperature coefficient of resistance.
• Semiconductors: In semiconductors, such as silicon, resistance decreases with temperature increase. This is due to the increased number of charge carriers available for conduction. At higher temperatures, more electrons in a semiconductor gain the energy needed to jump into the conduction band, reducing the material's resistance. This phenomenon leads to an increase in conductivity as temperature rises.

To find the magnitude of the charge on the dipole, we can use the formula for the torque experienced by a dipole in an electric field:

τ = pEsinθ

where:

• τ is the torque (4 Nm),
• p is the dipole moment,
• E is the electric field intensity (2×10⁵ N/C),
• θ is the angle between the dipole and the electric field (30°).

The dipole moment p is given by:

p = qd

where:

• q is the charge,
• d is the dipole length (2 cm = 0.02 m).

First, express the sine of the angle:

sin30° = 0.5

Substitute the known values into the torque formula:

4 = q × 0.02 × 2×10⁵ × 0.5

Simplify the equation:

4 = q × 0.01 × 2×10⁵

4 = q × 2000

Solve for q:

q = 4 / 2000 = 0.002 C = 2 mC

Therefore, the magnitude of the charge on the dipole is 2 mC.

The given integral equation is , which is an expression of Gauss's Law for electricity in integral form for electric flux through a closed surface. This equation indicates that the net electric flux (total number of electric field lines) through the closed surface is zero.

Let's evaluate the options:

The electric field inside the surface is necessarily uniform:
This is incorrect. A zero net electric flux through a surface doesn't imply uniformity of the electric field within that surface. The field can vary in different regions, maintaining an overall balance such that net flux remains zero.

The number of flux lines entering the surface must be equal to the number of flux lines leaving it:
Correct. A zero net electric flux indicates that whatever flux enters the surface is equally leaving it, ensuring the total entering equals the total leaving.

The magnitude of electric field on the surface is constant:
This is incorrect. The net flux equating to zero doesn't necessarily imply that the electric field's magnitude remains constant over the surface.

All the charges must necessarily be inside the surface:
This is incorrect. Having zero net electric flux doesn't require charges to be inside the surface. Charges can be outside, affecting the field in a way that flux entering equals flux leaving.

Thus, the correct interpretation of the given condition is that the number of flux lines entering the surface must be equal to the number of flux lines leaving it.