Chapter 22: Q. 51 (page 626)

FIGURE $P 22.51$ shows four charges at the corners of a square of side $L$ . What is the magnitude of the net force on $q$ ?

Short Answer

Expert verified

Magnitude of net force on charge is $(2 - \sqrt{2}) \frac{KQq}{L^{2}}$ .

Step by step solution

Figure for magnitude of the net force on charge

Figure is,

Step 2:

Columns law for force is,

$F_{C} = F_{1 on 2} = F_{2 on 1} = \frac{K |q_{1}| |q_{2}|}{r^{2}}$

$L$ is the distance between $q_{1}$ andlocalid="1649103824153" $q$ :

$F_{q_{1} on q} = \frac{{Kq}_{1} q_{\hat{i}}}{L^{2}}$

$\sqrt{2} L$ is the distance between $q_{2}$ andlocalid="1649103830714" $q$ :

$F_{q_{2} on q} = \frac{{Kq}_{q}}{(\sqrt{2} L)^{2}} \cos 45^{o} \hat{i} + \frac{{Kq}_{2} q}{(\sqrt{2} L)^{2}} \sin 45^{o} \hat{j}$

$= (\frac{{Kq}_{2} q}{(\sqrt{2} L)^{2}}) (\frac{1}{\sqrt{2}}) \hat{i} + (\frac{{Kq}_{2} q}{(\sqrt{2} L)^{2}}) (\frac{1}{\sqrt{2}}) \hat{j}$

$= (\frac{K (4 Q) q}{(\sqrt{2} L)^{2}}) (\frac{1}{\sqrt{2}}) \hat{i} + (\frac{K (4 Q) q}{(\sqrt{2} L)^{2}}) (\frac{1}{\sqrt{2}}) \hat{j}$

$= (\frac{4 KQq}{2 \sqrt{2} L^{2}}) \hat{i} + (\frac{4 KQq}{2 \sqrt{2} L^{2}}) \hat{j}$

Calculation for net force on charge

$L$ is distance for $q_{3}$ and $q$ :

$F_{q_{3} on q} = \frac{{Kq}_{3} q}{L^{2}} \hat{j}$

$= \frac{- KQq}{L^{2}} \hat{j}$

Net force is,

$F_{net} = F_{q_{1} on q} + F_{q_{2} on q} + F_{q_{3} on q}$

$= \frac{- KQq}{L^{2}} \hat{i} + (\frac{4 KQq}{2 \sqrt{2} L^{2}}) \hat{i} + (\frac{4 KQq}{2 \sqrt{2} L^{2}}) \hat{j} + \frac{- KQq}{L^{2}} \hat{j}$

$= \frac{KQq}{L^{2}} [- 1 + \frac{2}{\sqrt{2}}] \hat{i} + \frac{KQq}{L^{2}} [- 1 + \frac{2}{\sqrt{2}}] \hat{j}$

Magnitude of net force is,

$F_{net} = \sqrt{{(\frac{KQq}{L^{2}} [- 1 + \frac{2}{\sqrt{2}}])}^{2} + {(\frac{KQq}{L^{2}} [- 1 + \frac{2}{\sqrt{2}}])}^{2}}$

$= \sqrt{2} (\frac{KQq}{L^{2}} [- 1 + \frac{2}{\sqrt{2}}])$

$= (2 - \sqrt{2}) \frac{KQq}{L^{2}}$

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FIGURE $P 22.51$ shows four charges at the corners of a square of side $L$ . What is the magnitude of the net force on $q$ ?

Short Answer

Step by step solution

Figure for magnitude of the net force on charge

Step 2:

Calculation for net force on charge

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FIGURE P22.51shows four charges at the corners of a square of side L. What is the magnitude of the net force onq?

Short Answer

Step by step solution

Figure for magnitude of the net force on charge

Step 2:

Calculation for net force on charge

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Physical Quantities And Units

Thermodynamics

Work Energy and Power

Astrophysics

Oscillations

Conservation of Energy and Momentum

Study anywhere. Anytime. Across all devices.

FIGURE $P 22.51$ shows four charges at the corners of a square of side $L$ . What is the magnitude of the net force on $q$ ?