Answer to Question #311458 in Electric Circuits for Anne

Question

1. Two charges are located on the positive x-axis of a coordinate system. Charge 𝑞1 = 2 × 10^−9𝐶 is 2cm from the origin, and charge 𝑞2 = −3 × 10^−9𝐶 is 4cm from the origin. What is the total force exerted by these two charges on a charge 𝑞3 = 5 × 10^−9𝐶 located at the origin?

2. A negative charge of −0.50 × 10^−6𝐶 exerts a repulsive force of magnitude 0.20 N on an unknown charge 0.20 m away. What is the unknown charge (magnitude and sign)?

3. How many excess electrons must be placed on each of two small spheres spaced 3cm apart if the spheres are to have equal charge and if the force of repulsion between them is to be 10^−19𝑁?

4. What is the electric field 30cm from a charge 𝑞 = 4 × 10^−9𝐶?

Answer 1

Solution.

$1. q_1=2\sdot10^{-9}C;$

$q_2=-3\sdot10^{-9}C;$

$q_3=5\sdot10^{-9}C;$

$r_1=2cm=0.02m;$

$r_2=4cm=0.04m;$

$F_{13}=k \dfrac{q_1\sdot q_3}{r_1^2};$

$F_{13}=9\sdot10^9\dfrac{2\sdot10^{-9}\sdot5\sdot10^{-9}}{0.02^2}=2.25\sdot10^{-4}N;$

$F_{23}=k\dfrac{q_2\sdot q_3}{r_2^2};$

$F_{23}=9\sdot10^9\dfrac{3\sdot10^{-9}\sdot5\sdot10^{-9}}{0.04^2}=0.84\sdot10^{-4}N;$

$F=F_{13}-F_{23}=2.25\sdot 10^{-9} - 0.84\sdot 10^{-9}=1.41\sdot10^{-9}N;$

$2. q_1=-0.5\sdot10^{-6}C;$

$F=0.20N;$

$r=0.20m;$

$F=k\dfrac{q_1\sdot q_2}{r^2}\implies q_2=\dfrac{F\sdot r^2}{k\sdot q_1};$

$q_2=\dfrac{0.20\sdot0.20^2}{9\sdot10^9\sdot0.5\sdot 10^{-6}}=0.44\sdot10^{-6}C;$

$q_2=-0.44\sdot10^{-6}C;$

$3.$ $r=0.03m;$

$q_1=q_2=q;$

$F=10^{-19}N;$

$F=k\dfrac{q^2}{r^2}\implies q=\sqrt{\dfrac{F\sdot r^2}{k}};$

$q=\sqrt{\dfrac{10^{-19}\sdot0.03^2}{9\sdot10^9}}=1\sdot10^{-7}C;$

$q=Ne\implies N=\dfrac{q}{e}=\dfrac{1\sdot10^{-7}}{1.6\sdot10^{-19}}=625\sdot10^9;$

$4. r=0.3m;$

$q=4\sdot10^{-9}C;$

$E=\dfrac{F}{q};$

$E=k\dfrac{q}{r^2};$

$E=9\sdot10^9\dfrac{4\sdot10^{-9}}{0.09}=400V/m;$

Answer:$1.F=1.41\sdot10^{-9}N;$

$2. q_2=-0.44\sdot10^{-6}C;$

$3.$ $N=625\sdot10^9;$

$4. E=400 V/m;$