Explain axial field or electric field on the axial line of an electric dipole i.e. end on position

A line passing through the positive and negative charges of the dipole is called the axial line of the electric dipole. Consider a point P on the axial line of a dipole (situated in a vacuum) of length 2a at a distance r from the midpoint O.

Electric intensity at P due to –q charge at A, i.e.,

\({{E}_{1}}=k\frac{q}{A{{P}^{2}}}=k\frac{q}{{{\left( r+a \right)}^{2}}}\)

It is represented in magnitude and direction by \(\overrightarrow{PQ}\), i.e., \(\overrightarrow{{{E}_{1}}.}\) Electric intensity at P due to +q charge at B, i.e.,

\({{E}_{2}}=k\frac{q}{B{{P}^{2}}}=k\frac{q}{{{\left( r-a \right)}^{2}}}\)

It is represented in magnitude and direction by \(\overrightarrow{PR},\) i.e., \(\overrightarrow{{{E}_{2}}.}\) Let \(\overrightarrow{E}\) be the resultant electric intensity at P.

According to the principle of superposition of electric fields, \(\overrightarrow{E}=\overrightarrow{{{E}_{1}}}+\overrightarrow{{{E}_{2}}}\)

Since \(\overrightarrow{{{E}_{1}}}\) and E2 act in opposite directions, the resultant electric intensity \(\left( \overrightarrow{E} \right)\) at P due to the dipole is represented by \(\overrightarrow{PS}.\) Clearly, as \(\left| \overrightarrow{{{E}_{2}}} \right|>\left| \overrightarrow{{{E}_{1}}} \right|,\) \(E={{E}_{2}}-{{E}_{1}}=k\left[ \frac{q}{{{\left( r-a \right)}^{2}}}-\frac{q}{{{\left( r+a \right)}^{2}}} \right]\) \(=kq\left[ \frac{{{\left( r+a \right)}^{2}}-{{\left( r-a \right)}^{2}}}{{{\left( r-a \right)}^{2}}{{\left( r+a \right)}^{2}}} \right]=kq\frac{4ra}{{{\left( {{r}^{2}}-{{a}^{2}} \right)}^{2}}}\)

Or \(E=k\frac{2\times 2qa\times r}{{{\left( {{r}^{2}}-{{a}^{2}} \right)}^{2}}}=k\frac{2pr}{{{\left( {{r}^{2}}-{{a}^{2}} \right)}^{2}}}\)

Since \(\overrightarrow{E}\,and\,\overrightarrow{p}\) are in the same direction,

\(\overrightarrow{E}=k\frac{2\overrightarrow{p}\,r}{{{\left( {{r}^{2}}-{{a}^{2}} \right)}^{2}}}=\frac{1}{4\pi {{\in }_{0}}}\frac{2\overrightarrow{p}\,r}{{{\left( {{r}^{2}}-{{a}^{2}} \right)}^{2}}}\) … (1)

Special cases:

  1. If r is very large as compared to a, i.e., if the dipole is short, a2 can be neglected as compared to r2 and as such,
\(\overrightarrow{E}=k\frac{2\,\overrightarrow{p}}{{{r}^{3}}}=\frac{1}{4\pi \,{{\in }_{0}}}\frac{2\overrightarrow{p}}{{{r}^{3}}}\) … (2)

  1. E varies as \(1/{{r}^{3}}\) in case of an electric dipole whereas in case of monopole, \(E\propto \frac{1}{r}.\)
  2. If the electric dipole is situated in a medium of relative permittivity \({{\in }_{r}},\) then
\(\overrightarrow{E}=k\frac{2\overrightarrow{p}\,r}{{{\in }_{r}}{{\left( {{r}^{2}}-{{a}^{2}} \right)}^{2}}}=\frac{1}{4\pi {{\in }_{0}}{{\in }_{r}}}\frac{2\overrightarrow{p}\,r}{{{\left( {{r}^{2}}-{{a}^{2}} \right)}^{2}}}\)

and \(\overrightarrow{E}=k\frac{2\overrightarrow{p}\,r}{{{\in }_{r}}{{r}^{3}}}=\frac{1}{4\pi {{\in }_{0}}{{\in }_{r}}}\frac{2\overrightarrow{p}}{{{r}^{3}}}\) (Short dipole) … (3)

1 Comment

  1. Khlainkupar Lyngkhoi

    Good

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