A charge of 3.0 nC and a charge of 5.8 nC are separated by 53.40 cm. Find the equilibrium position for a -4.7 nC charge.
Since q1 and q2 are positive charges, q3 in its equilibrium position has to be between them at the distance “x” from the charge q1.
F1=F2.
Use Coulomb’s Law
F1= k•q1•q3/x²,
F2 = k •q2•q3/(r-x)²,
k•q1•q3/x²= k •q2•q3/(r-x)²,
q1 /x²= q2 /(r-x)².
Solve for “x”
To find the equilibrium position for a -4.7 nC charge, we can use Coulomb's Law and set the forces between the charges equal to each other. Coulomb's Law states that the force between two point charges is given by:
F = (k * |q1 * q2|) / r^2
Where:
- F is the force between the charges
- k is the electrostatic constant, k = 8.99 x 10^9 Nm^2/C^2
- q1 and q2 are the magnitudes of the charges
- r is the separation between the charges
Let's denote the equilibrium position we want to find as x, where the negative charge (-4.7 nC) is a distance x away from the 3.0 nC charge, and (53.40 cm - x) away from the 5.8 nC charge.
Using Coulomb's Law, the force between the -4.7 nC charge and the 3.0 nC charge is:
F1 = (k * |q1 * q2|) / r^2
F1 = (8.99 x 10^9 Nm^2/C^2) * (4.7 x 10^-9 C) * (3.0 x 10^-9 C) / x^2
Similarly, the force between the -4.7 nC charge and the 5.8 nC charge is:
F2 = (k * |q1 * q2|) / r^2
F2 = (8.99 x 10^9 Nm^2/C^2) * (4.7 x 10^-9 C) * (5.8 x 10^-9 C) / (53.40 cm - x)^2
Since the equilibrium position is where the forces are equal, we can set F1 equal to F2:
(8.99 x 10^9 Nm^2/C^2) * (4.7 x 10^-9 C) * (3.0 x 10^-9 C) / x^2 = (8.99 x 10^9 Nm^2/C^2) * (4.7 x 10^-9 C) * (5.8 x 10^-9 C) / (53.40 cm - x)^2
Simplifying and solving for x:
(3.0 x 10^-9 C) / x^2 = (5.8 x 10^-9 C) / (53.40 cm - x)^2
Cross-multiplying:
(3.0 x 10^-9 C) * (53.40 cm - x)^2 = (5.8 x 10^-9 C) * x^2
Expanding and rearranging:
(3.0 x 10^-9 C) * (2851.16 cm^2 - 106.8 cm * x + x^2) = (5.8 x 10^-9 C) * x^2
(8553.48 x 10^-9 C cm^2 - 320.4 x 10^-9 C cm * x + 3.0 x 10^-9 C cm^2 * x^2) = (5.8 x 10^-9 C) * x^2
Subtracting (5.8 x 10^-9 C) * x^2 from both sides:
(8553.48 x 10^-9 C cm^2 - 320.4 x 10^-9 C cm * x) = (2.8 x 10^-9 C) * x^2
Rearranging and factoring:
(3.0 x 10^-9 C) * x^2 + (320.4 x 10^-9 C cm) * x - (8553.48 x 10^-9 C cm^2) = 0
Now, we have a quadratic equation in terms of x. We can solve it using the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
Where:
- a = (3.0 x 10^-9 C)
- b = (320.4 x 10^-9 C cm)
- c = (-8553.48 x 10^-9 C cm^2)
Plugging in the values and solving for x will give us the equilibrium position for the -4.7 nC charge.