Six point charges are kept apart from each other on the circumference of a circle of radius R as shown in figure. The net electric field at the center of the circle is ______. ( is permittivity of free space) [2026]

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Q.
Six point charges are kept $60 °$ apart from each other on the circumference of a circle of radius R as shown in figure. The net electric field at the center of the circle is ______.

( $ε_{0}$ is permittivity of free space) [2026]

1 $- (\frac{5 Q}{8 π ε_{0} R^{2}}) (\hat{i} - 3 \hat{j})$

2 $\frac{Q}{4 π ε_{0} R^{2}} (\sqrt{3} \hat{i} - \hat{j})$

3 $- \frac{5 Q}{8 π ε_{0} R^{2}} (\hat{i} + \sqrt{3} \hat{j})$

4 $- \frac{Q}{4 π ε_{0} R^{2}} (\sqrt{3} \hat{i} - \hat{j})$

Ans.
(4)

$Let \frac{k Q}{r^{2}} = E_{0}$

${\vec{E}}_{net} = 2 E_{0} \cos 30 ° (- \hat{i}) + 2 E_{0} \sin 30 ° (\hat{j})$

$= \frac{2 k Q}{r^{2}} [\frac{\sqrt{3}}{2} (- \hat{i}) + \frac{1}{2} \hat{j}]$

$= \frac{- Q}{4 π ε_{0} r^{2}} (\sqrt{3} \hat{i} - \hat{j})$

Solution

Q.
Six point charges are kept $60 °$ apart from each other on the circumference of a circle of radius R as shown in figure. The net electric field at the center of the circle is ______.

( $ε_{0}$ is permittivity of free space) [2026]

1 $- (\frac{5 Q}{8 π ε_{0} R^{2}}) (\hat{i} - 3 \hat{j})$

2 $\frac{Q}{4 π ε_{0} R^{2}} (\sqrt{3} \hat{i} - \hat{j})$

3 $- \frac{5 Q}{8 π ε_{0} R^{2}} (\hat{i} + \sqrt{3} \hat{j})$

4 $- \frac{Q}{4 π ε_{0} R^{2}} (\sqrt{3} \hat{i} - \hat{j})$

Ans.
(4)

$Let \frac{k Q}{r^{2}} = E_{0}$

${\vec{E}}_{net} = 2 E_{0} \cos 30 ° (- \hat{i}) + 2 E_{0} \sin 30 ° (\hat{j})$

$= \frac{2 k Q}{r^{2}} [\frac{\sqrt{3}}{2} (- \hat{i}) + \frac{1}{2} \hat{j}]$

$= \frac{- Q}{4 π ε_{0} r^{2}} (\sqrt{3} \hat{i} - \hat{j})$

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Solution

Q. Six point charges are kept 60° apart from each other on the circumference of a circle of radius R as shown in figure. The net electric field at the center of the circle is ______. (ε0 is permittivity of free space) [2026]

1 -(5Q8πε0R2)(i^-3j^)

2 Q4πε0R2(3i^-j^)

3 -5Q8πε0R2(i^+3j^)

4 -Q4πε0R2(3i^-j^)

Ans. (4) Let kQr2=E0 E→net=2E0cos30°(-i^)+2E0sin30°(j^) =2kQr2[32(-i^)+12j^] =-Q4πε0r2(3 i^-j^)

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Q.
Six point charges are kept $60 °$ apart from each other on the circumference of a circle of radius R as shown in figure. The net electric field at the center of the circle is ______.

( $ε_{0}$ is permittivity of free space) [2026]

1 $- (\frac{5 Q}{8 π ε_{0} R^{2}}) (\hat{i} - 3 \hat{j})$

2 $\frac{Q}{4 π ε_{0} R^{2}} (\sqrt{3} \hat{i} - \hat{j})$

3 $- \frac{5 Q}{8 π ε_{0} R^{2}} (\hat{i} + \sqrt{3} \hat{j})$

4 $- \frac{Q}{4 π ε_{0} R^{2}} (\sqrt{3} \hat{i} - \hat{j})$

Ans.
(4)

$Let \frac{k Q}{r^{2}} = E_{0}$

${\vec{E}}_{net} = 2 E_{0} \cos 30 ° (- \hat{i}) + 2 E_{0} \sin 30 ° (\hat{j})$

$= \frac{2 k Q}{r^{2}} [\frac{\sqrt{3}}{2} (- \hat{i}) + \frac{1}{2} \hat{j}]$

$= \frac{- Q}{4 π ε_{0} r^{2}} (\sqrt{3} \hat{i} - \hat{j})$