To determine the deficiency of electrons for a body with a charge of

$10^{-12} \, \text{C}$

, follow these steps:

1. Charge of a single electron: The charge of one electron is
   $e = 1.6 \times 10^{-19} \, \text{C}$ 

   .

2. Number of electrons (deficiency): The total number of electrons causing the charge is given by: 

   $$n = \frac{\text{Total charge}}{\text{Charge of one electron}} = \frac{10^{-12}}{1.6 \times 10^{-19}} = 6.25 \times 10^{6}.$$ 

3. Interpretation: Since the body is positively charged, it has a deficiency of
   $6.25 \times 10^6$ 

   electrons.

When a body is charged by rubbing, its weight may increase or decrease slightly because:

1. Gain of electrons (negative charge):
   • If the body gains electrons during the process, it becomes negatively charged.
   • Since electrons have mass (
     $9.1 \times 10^{-31} \, \text{kg}$ 

     ), the body’s weight will increase slightly due to the addition of electrons.

2. Loss of electrons (positive charge):
   • If the body loses electrons during rubbing, it becomes positively charged.
   • The loss of electrons reduces the body's total mass, so its weight will decrease slightly.

Thus, depending on whether the body gains or loses electrons, its weight may increase or decrease slightly.

The dielectric constant (also called the relative permittivity,

$\varepsilon_r$

) is a dimensionless quantity because it is defined as the ratio:

$$\varepsilon_r = \frac{\varepsilon}{\varepsilon_0}$$

Where:

• 
  $\varepsilon$ 

  = permittivity of the medium (units: $\text{F/m}$ 

  , farads per meter)

• 
  $\varepsilon_0$ 

  = permittivity of free space ( $\text{F/m}$ 

  , farads per meter)

Since it is a ratio of two quantities with the same unit, the dielectric constant has no unit (it is dimensionless).

The correct statement is "the total charge of the universe is constant" because:

• Conservation of charge is a fundamental principle in physics. It states that the total electric charge in an isolated system remains constant, regardless of the processes occurring within the system.
• This principle applies to all known interactions and processes, such as chemical reactions, particle collisions, and even cosmic events, meaning that charge cannot be created or destroyed—only transferred between objects.

Thus, the total charge in the universe remains constant.

The electric charge in accelerated motion produces electromagnetic radiation.

Here’s why:

• When a charged particle accelerates (changes velocity), it disturbs the surrounding electromagnetic field. These disturbances propagate as electromagnetic waves, which include light, radio waves, X-rays, etc.
• This is a fundamental result of Maxwell's equations, which describe how electric and magnetic fields interact with charged particles.

Thus, an accelerating electric charge produces electromagnetic radiation, which can be detected as waves of energy moving through space.

When a soap bubble is given a negative charge, the radius of the bubble increases. Here's why:

• A soap bubble with charge experiences electrostatic repulsion between the like charges distributed on its surface. This repulsion causes the surface to expand.
• The electrostatic pressure
  $P$ 

  on the surface of a charged sphere (or bubble) is given by the formula:

$$P = \frac{2 \sigma^2}{\varepsilon_0}$$

where

$\sigma$

is the surface charge density, and

$\varepsilon_0$

is the permittivity of free space. This pressure acts to push the bubble outward.

• To counteract this, the bubble expands, increasing its radius in response to the electrostatic repulsion.

Thus, when a soap bubble is negatively charged, the repulsion between the charges causes the bubble's radius to increase.

When the two metallic spheres are close to each other, their charges induce redistribution of charges on their surfaces due to proximity. This redistribution impacts the force of interaction:

• For like charges (both positive or both negative): The redistribution of charge reduces the repulsive force because the induced charges create opposing electric fields, weakening the overall repulsion.
• For unlike charges (one positive, one negative): The redistribution enhances the attractive force because the induced charges increase the local electric field, strengthening the attraction.

Thus, the force of interaction is greater for unlike charges due to this enhancement caused by charge redistribution.

The statement "The charge without mass is not possible" is correct based on our current understanding of physics because:

• All known charged particles, such as electrons, protons, and ions, have mass associated with them. There is no experimental evidence of a physical entity that carries charge but has zero mass.
• Even in the case of hypothetical particles like the photon, which is massless, it does not carry electric charge. Charged particles inherently possess mass due to their nature in both classical and quantum physics.

Thus, charge is always associated with some mass in all observed phenomena.

The relativistic formula for mass,

$$m = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}}$$

accounts for how mass increases with velocity. This behavior arises because mass is a form of energy, and energy is affected by motion under relativity.

However, electric charge (

$q$

) is invariant under relativistic mechanics. Charge does not depend on the velocity of the particle. It remains constant in all inertial reference frames, which is a fundamental principle in physics.

Thus, the equivalent relation for electric charge is simply:

$$q = q_0$$

This reflects the fact that charge does not vary with velocity, unlike mass.