Two point charges are located on the x-axis as follows: charge 1 = 4 is located at = −2 and 2 = −2 is located at = 2 . If a test charge = 2 is placed at = 2 .At what point between two like charges of equal magnitude separated by a distance would the electric field be equal to zero.
Answers
Answer-The work that must be done on q_3q
3
by an external force \displaystyle \vec{F}_{ext}
F
ext
is equal to the difference between two quantities: the potential energy U associated with q_3q
3
when it is at x = 2a and the potential energy when it is infinitely far away.
The second of these two is zero, so the work that must be done is equal to U. The distances between the charges are r_{13}=2ar
13
=2a and r_{22} = ar
22
=a, so
\displaystyle W = U = \frac{q_3}{4 \pi \varepsilon_0} \left( \frac{q_1}{r_{13}} + \frac{q_2}{r_{23}}\right)W=U=
4πε
0
q
3
(
r
13
q
1
+
r
23
q
2
)
\displaystyle = \frac{+e}{4 \pi\varepsilon_0} \left( \frac{-e}{2a} +\frac {e}{a}\right) = \frac{+e^2}{8 \pi \varepsilon_0a}=
4πε
0
+e
(
2a
−e
+
a
e
)=
8πε
0
a
+e
2