Given:
<span>
</span>q1 = q<span>
</span>q2 = 4q<span>
</span>q3 = q<span>
</span>d = 2.00 cm<span>
</span>xq1 = 0 cm<span>
</span>xq2 = 2.00 cm<span>
</span>q = 2.00 nC
Let us determine the forces:<span>
</span>F<span>1on3 </span>= K ⋅ |q1<span>| ⋅ |q</span>3| / r2<span> = K ⋅ |q|
⋅ |q| / (d</span>1-3)2<span> = K ⋅ q</span>2 / (d1-3)2<span>
</span>F<span>2on3 </span>= K ⋅ |q2<span>| ⋅ |q</span>3| / r2<span> = K ⋅ |4q|
⋅ |q| / (d</span>3-2)2<span> = K ⋅ 4q</span>2 / (d3-2)2
We're able to solve for the
variables in both cases since the forces are supposed to equal each
other in each case.
F<span>1on3 </span>= F2on3<span>
</span><span>K ⋅ q</span>2 / x<span>2 </span>= K ⋅ 4q2 / (2.00 cm - x)2<span>
</span>1/x2 = 4/ [4.00 cm2<span> - (4.00 cm⋅x)
+ x</span>2]<span>
</span>[4.00 cm2<span> - (4.00 cm ⋅ x)
+ x</span>2] / x2 = 4<span>
</span>4.00 cm<span>2 </span>/ x2<span> - (4.00 cm ⋅ x)
/ x</span>2 + x<span>2 </span>/ x2 - 4 = 0<span>
</span>4.00 cm<span>2 </span>/ x2 - (4.00 cm) / x + 1 - 4 = 0<span>
</span>4.00 cm<span>2 </span>/ x2 - (4.00 cm) /x - 3 = 0<span>
</span>3 = 4.00 cm<span>2 </span>/ x2 - (4.00 cm) / x<span>
</span>3x<span>2 </span>= 4.00 cm - 4.00 cm ⋅ x<span>
</span>3x2<span> + 4.00 cm ⋅ x
- 4.00 cm = 0</span>
<span>
</span><span>After we plug that into the quadratic equation, we get:<span>
</span>x = -2 & x = 2/3 <span>
</span><span>x3,r, x3,ℓ<span> = 0.667,-2.00 cm, cm</span></span></span>
