Question

physical significance of displacement vector.

Answer 1

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In physics, the electric displacement field, denoted by D, is a vector field that appears in Maxwell's equations. It accounts for the effects of free and bound charge within materials. "D" stands for "displacement", as in the related concept of displacement current in dielectrics. In free space, the electric displacement field is equivalent to flux density, a concept that lends understanding to Gauss's law. In SI, it is expressed in units of coulomb per metre squared (C⋅m−2).

DefinitionEdit

In a dielectric material, the presence of an electric field E causes the bound charges in the material (atomic nuclei and their electrons) to slightly separate, inducing a local electric dipole moment. The electric displacement field D is defined as



where is the vacuum permittivity (also called permittivity of free space), and P is the (macroscopic) density of the permanent and induced electric dipole moments in the material, called the polarization density.

The displacement field satisfies Gauss's lawin a dielectric:

Proof

Separate the total volume charge density into free and bound charges:

The density can be rewritten as a function of the polarization P:

The polarization P is defined to be a vector field whose divergence yields the density of bound charges ρb in the material. The electric field satisfies the equation:

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Answer 2

The physical significance of the displacement vector is that it tells that how much a particle or an object displaced with its direction.

Explanation:

• Displacement can be defined as the minimum distance between two points along with the direction.
• Suppose a point particle is initially at a point $\[{P_1}\]$ and its position vector is $\overrightarrow {{r_1}}$.
• After some time, it goes to the point ${P_2}$ and its position vector becomes $\overrightarrow {{r_2}}$.
• Then, the displacement vector mathematically defined as:

       $\Delta \overrightarrow r = \overrightarrow {{r_2}} - \overrightarrow {{r_1}}$

• Therefore, the magnitude of $\Delta \overrightarrow r$ gives how much particle is displaced from its initial position to the final position.
• The unit vector along this displacement vector gives the direction of displacement.
• For example, an object moves from point A to point B in time $t$.
• If position vectors at points A  and B are given as:

        $\[\begin{gathered} A = {{\vec r}_A} = 5\hat i + 3\hat j + 4\hat k \hfill \\ B = {{\vec r}_B} = 2\hat i + 2\hat j + 1\hat k \hfill \\ \end{gathered} \]$

• Therefore, the displacement vector is given as:

        $\[\begin{gathered} \Delta \vec r = {{\vec r}_B} - {{\vec r}_B} \hfill \\ \Delta \vec r = - 3\hat i - \hat j - 3\hat k \hfill \\ \end{gathered} \]$

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