Two point charges 1.5 cm apart have an electric potential energy -150 mu J. The total charge is 34 nC. what are q_1 and q_2?
PE has to be referenced to a particular point. Where was the 150E-6 J measured?
To find the values of q₁ and q₂, we can use the equation for electric potential energy:
U = k(q₁q₂/r)
Where:
U is the electric potential energy
k is the electrostatic constant (9 × 10^9 Nm²/C²)
q₁ and q₂ are the charges
r is the separation between the charges
In this case, we are given:
U = -150 μJ = -150 × 10^-6 J (Note: μ = 10^-6)
r = 1.5 cm = 0.015 m
The total charge is given as 34 nC, so q₁ + q₂ = 34 nC = 34 × 10^-9 C
Let's solve for q₁ and q₂ step by step:
Step 1: Expressing the potential energy in terms of Coulombs:
U = k(q₁q₂/r)
-150 × 10^-6 = (9 × 10^9) (q₁q₂/0.015)
Step 2: Rearranging the equation:
(q₁q₂/0.015) = -150 × 10^-6 / (9 × 10^9)
Step 3: Simplifying the equation:
(q₁q₂/0.015) = -1.67 × 10^-14
Step 4: Substituting the value of the total charge:
(q₁ × (34 × 10^-9 - q₁)/0.015) = -1.67 × 10^-14
Step 5: Expanding and rearranging the equation:
34q₁ × 10^-9 - q₁² = -1.67 × 10^-14 × 0.015
Step 6: Simplifying and rearranging further:
34q₁ - q₁² × 10^9 = -1.67 × 10^-16
Step 7: Moving all terms to one side of the equation:
q₁² × 10^9 - 34q₁ - 1.67 × 10^-16 = 0
Step 8: Solving for q₁ using the quadratic formula:
q₁ = [-(-34) ± √((-34)² - 4 × 10^9 × (-1.67 × 10^-16))] / (2 × 10^9)
By solving the quadratic equation, we can find the values of q₁ and q₂.