A +5.7 C and a -3.5 C charge are placed 25 cm apart. Where can a third charge be placed so that it experiences no net force? (Restrict your analysis to a linear one.)
To find the point where a third charge experiences no net force, we need to calculate the distance from both the +5.7 μC charge and the -3.5 μC charge.
Let's assume the third charge is q3 and its distance from the +5.7 μC charge is x. Since the distance between the charges is given as 25 cm, the distance from the -3.5 μC charge to q3 would be (25 - x) cm.
According to Coulomb's Law, the electric force between two charges is given by:
F = k * (q1 * q2) / r^2
Where:
- F is the electric force
- k is Coulomb's constant (9 × 10^9 N m^2 / C^2)
- q1 and q2 are the two charges
- r is the distance between the charges
For the third charge to experience no net force, the magnitudes of the forces exerted by the two charges must be equal. Mathematically, this can be written as:
k * (q1 * q3) / x^2 = k * (q2 * q3) / (25 - x)^2
By canceling out k * q3 from both sides of the equation, and assuming q1 = +5.7 μC and q2 = -3.5 μC, we can simplify the equation as follows:
q1 / x^2 = q2 / (25 - x)^2
Now we can solve this equation algebraically to find the value of x.
Multiplying both sides by x^2 * (25 - x)^2, we get:
q1 * (25 - x)^2 = q2 * x^2
Substituting the values for q1, q2, and rearranging the equation, we have:
(5.7 μC) * (25 - x)^2 = (-3.5 μC) * x^2
Expanding both sides of the equation and simplifying, we get:
(5.7 μC) * (625 - 50x + x^2) = (-3.5 μC) * x^2
Now, let's solve for x:
(5.7 μC) * (625 - 50x + x^2) = (-3.5 μC) * x^2
Distributing and rearranging, we have:
3575 μC - 285 μC * x + 5.7 μC * x^2 = -3.5 μC * x^2
Bringing all terms to one side of the equation:
0 = -3.5 μC * x^2 - 5.7 μC * x^2 - 285 μC * x + 3575 μC
Combining like terms, we get:
0 = -9.2 μC * x^2 - 285 μC * x + 3575 μC
This quadratic equation can be solved using various methods such as factoring, completing the square, or using the quadratic formula. Once calculated, the values of x can be used to determine the distances at which the third charge experiences no net force from the other charges.