To calculate the susceptibility, we use the definition of bound charge density in terms of polarization (P) and susceptibility (chi), alongside the relationship with electric field (E). When in vacuum or air, the electric displacement field (D), the electric field (E), and the polarization (P) are related as follows:

[D = varepsilon_0 E + P]

The polarization (P) can also be expressed as:

[P = chi varepsilon_0 E]

The bound charge density (rho_b) is related to the polarization by the equation:

[nabla cdot P = -rho_b]

For a uniform polarization, (rho_b) can simply be equated to the volume density of bound charges, which is given as 12 units in the question.

However, the information provided doesn’t directly relate to how we usually calculate susceptibility. Susceptibility (chi) is a measure of how much a material will become polarized in an external electric field, affecting its polarization (P), but the relationship to bound charge density (rho_b) and free charge density requires additional context about the electric field (E) or the medium’s permittivity.

Given only the bound charge density (12 units) and free charge density (6 units), and without information about the electric field (E) or the material’s permittivity (varepsilon), the

To calculate the energy in an electric field, we first need to understand what you mean by “flux density” and “field intensity” in the context of energy calculation. However, one common formula used to calculate energy density (u) in an electric field is given by the equation:

[ u = frac{1}{2} epsilon E^2 ]

where:

– ( u ) is the energy density (energy per unit volume, J/m(^3)),

– (epsilon) is the permittivity of the medium (in vacuum, (epsilon_0 = 8.85 times 10^{-12}) F/m),

– (E) is the electric field intensity (V/m).

Since you’ve provided “flux density” and “field intensity” without specifying units or distinguishing between these terms in a standard context (flux density might imply charge density or magnetic flux density, and field intensity usually refers to electric field intensity), I’ll assume you’re referring to the electric field intensity ((E)) by “field intensity” and possibly referring to a related concept aligned with permittivity ((epsilon)) or electric displacement field ((D)) by “flux density.” However, this interpretation might not perfectly match your parameters without more precise definitions and units.

If “6 units” for flux density refer to the electric displacement field ((D)) and “4 units” for field intensity refers to the electric field ((E

To find the distance between the midpoint of the two charges and the point P, we first understand that the charges are separated by 1 cm, implying they lie on a straight line with each other. The charges are 3C and -3C, but this information is not directly relevant for calculating the distance to the midpoint; it is more relevant in electrostatic force or field calculations.

Given distances from P:

– The distance from P to the 3C charge is 5 cm.

– The distance from P to the -3C charge is 7 cm.

Since the charges are 1 cm apart, if we represent this linearly, we have two scenarios depending on the order of the charges regarding point P:

### 1. P closer to 3C charge:

– P —– 5cm —– 3C —– 1cm —– -3C —– 7cm (not possible based on the given distances; this arrangement would mean P is closer to -3C)

### 2. P closer to -3C charge, which fits the given distances:

– 3C —– 1cm —– -3C —– 5cm —– P —– 7cm (also not logical based on initial direct reading but helps picturing the scenario opposite to the initial assumptions)

### Correct logical arrangement:

– Considering the midpoint between charges is directly in the middle of the 1 cm gap, making the distances from P to each charge conceptually misunderstood in the depiction.

To find the angle at which the potential due to a dipole is measured, given the distances from the charges, consider a dipole consisting of charges (+q) and (-q), separated by a distance (2a). The position where the potential is being measured forms a triangle with the line joining the charges. Given that the distances from the positive and negative charges are 12 cm and 11 cm, respectively, and the separation between the charges ((2a)) is 2 cm, we can find the angle using the law of cosines.

Let us denote:

– (r_+) as the distance from the positive charge ((12 cm)),

– (r_-) as the distance from the negative charge ((11 cm)),

– (d = 2 cm) as the distance between the charges,

– (theta) as the angle the position vector (from the negative to the positive charge) makes with the perpendicular bisector of the dipole.

Using the law of cosines on the triangle formed by the position of measurement and the two charges, we have:

[r_+^2 = r_-^2 + d^2 – 2 cdot r_- cdot d cdot cos(theta)]

Substituting values:

[12^2 = 11^2 + 2^2 – 2 cdot 11 cdot 2 cdot cos(theta)]

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