Answer

Verified

446.1k+ views

**Hint:**We will use Coulomb’s law equation to derive an expression for the intensity of the electric field. The electric fields will be calculated at a point with different distances. the expression will be the difference of these electric fields being calculated.

**Formula used:**

\[E=\dfrac{1}{4\pi {{\varepsilon }_{0}}}\dfrac{q}{{{r}^{2}}}\]

**Complete step-by-step answer:**

A line passing through the pair of equal and opposite charges of a dipole is called the axial line.

Consider a diagram representing the axial line of a dipole.

Let the charges of a dipole be represented by \[-q\]and \[+q\]. Let the distance between the charges be \[2a\]. Let ‘P’ be a point on the axis line at which the electric field to be determined.

Consider the figure while going through the following steps.

The electric field at a point P with the distance BP is expressed as follows.

Along the line BP due to the charge\[+q\].

\[\begin{align}

& {{E}_{B}}=\dfrac{1}{4\pi {{\varepsilon }_{0}}}\dfrac{q}{{{(BP)}^{2}}} \\

& \Rightarrow {{E}_{B}}=\dfrac{1}{4\pi {{\varepsilon }_{0}}}\dfrac{q}{{{(r-a)}^{2}}} \\

\end{align}\]

The electric field at a point P with the distance PA is expressed as follows.

Along the line PA due to the charge \[-q\].

\[\begin{align}

& {{E}_{A}}=\dfrac{1}{4\pi {{\varepsilon }_{0}}}\dfrac{q}{{{(AP)}^{2}}} \\

& \Rightarrow {{E}_{A}}=\dfrac{1}{4\pi {{\varepsilon }_{0}}}\dfrac{q}{{{(r+a)}^{2}}} \\

\end{align}\]

The net field at the point P is calculated as the difference between the electric fields at the distances BP and PA.

Thus, the net electric field at point P is,

\[{{E}_{P}}={{E}_{B}}-{{E}_{A}}\]

Substitute the expressions of the electric fields at the distances BP and PA in the above equation.

\[{{E}_{P}}=\dfrac{1}{4\pi {{\varepsilon }_{0}}}\dfrac{q}{{{(r-a)}^{2}}}-\dfrac{1}{4\pi {{\varepsilon }_{0}}}\dfrac{q}{{{(r+a)}^{2}}}\]

Take the common terms aside and solve further.

\[{{E}_{P}}=\dfrac{q}{4\pi {{\varepsilon }_{0}}}\left( \dfrac{1}{{{(r-a)}^{2}}}-\dfrac{1}{{{(r+a)}^{2}}} \right)\]

Continue the further calculation.

\[\begin{align}

& {{E}_{P}}=\dfrac{q}{4\pi {{\varepsilon }_{0}}}\left( \dfrac{4ar}{{{({{r}^{2}}-{{a}^{2}})}^{2}}} \right) \\

& \Rightarrow {{E}_{P}}=\dfrac{2qa}{4\pi {{\varepsilon }_{0}}}\left( \dfrac{2a}{{{({{r}^{2}}-{{a}^{2}})}^{2}}} \right) \\

\end{align}\]

From the diagram it’s clear that, \[2aq=p\]and the constant value \[{}^{1}/{}_{4\pi {{\varepsilon }_{0}}}\]can be expressed using a variable ‘k’.

So, we have the expression for the electric field at point P along the line BP as follows.

\[{{E}_{P}}=\dfrac{2kpr}{{{({{r}^{2}}-{{a}^{2}})}^{2}}}\]

Consider a special case, for which the condition is, \[2a << r\], the expression for the electric field can be derived as,

\[{{E}_{P}}=\dfrac{2kp}{{{r}^{3}}}\]

Therefore, the expression for the intensity of the electric field produced by an electric dipole at a point on its axial line is \[{{E}_{P}}=\dfrac{2kp}{{{r}^{3}}}\].

**Note:**The formula used to derive the expression is Coulomb’s law for a point charge. This is because we will be solving the electric field at a point, but considering two different distances. If the electric field is given to be within the distance between the charges of a dipole, then, the final expression should be used as discussed above.

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

A rainbow has circular shape because A The earth is class 11 physics CBSE

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

How do you graph the function fx 4x class 9 maths CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Why is there a time difference of about 5 hours between class 10 social science CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell