A positive charge q₁ = 1.00 μC is fixed at the origin, and a second charge q2=-2.00 μC is fixed at x = 10.0 cm. Where along the x-axis should a third charge be positioned so that it experiences no force?
To find the position along the x-axis where a third charge experiences no force, we can use Coulomb's Law:
F = k * q₁ * q₂ / r²
Here, F is the force between q₁ and q₂, k is the electrostatic constant (9.0 x 10^9 N m²/C²), q₁ is the charge at the origin (1.00 μC = 1.00 x 10^-6 C), q₂ is the charge at x = 10.0 cm (-2.00 μC = -2.00 x 10^-6 C), and r is the distance between q₁ and q₂.
Since we want the third charge to experience no force, the force between the third charge and q₂ should be zero:
F = k * q₂ * q₃ / (10.0 cm - x)² = 0
We can solve this equation to find the value of x for which the force is zero:
k * q₂ * q₃ / (10.0 cm - x)² = 0
q₂ * q₃ / (10.0 cm - x)² = 0
Since the product q₂ * q₃ cannot be zero (as q₂ = -2.00 μC and q₃ is unknown), the denominator (10.0 cm - x)² must be zero. This occurs when x = 10.0 cm.
Therefore, the third charge should be positioned at x = 10.0 cm along the x-axis in order to experience no force.
To find the position along the x-axis where a third charge experiences no force, we can use the concept of electric forces and Coulomb's law.
Let's assume the position of the third charge q₃ along the x-axis is x meters.
The electric force between two charges can be calculated using Coulomb's law:
F = k * |q₁ * q₂| / r²
where:
F is the electric force between charges q₁ and q₂
k is the electrostatic constant (k = 9.0 x 10^9 N m²/C²)
| | represents taking the absolute value
r is the distance between the charges
In this problem, we want the third charge q₃ to experience no force. It means that the net force on q₃ should be zero. Mathematically, it can be expressed as:
F₁₃ + F₂₃ = 0
where:
F₁₃ is the force between charges q₁ and q₃
F₂₃ is the force between charges q₂ and q₃
Let's calculate these forces separately.
1. Force between q₁ and q₃:
F₁₃ = k * |q₁ * q₃| / r₁₃²
2. Force between q₂ and q₃:
F₂₃ = k * |q₂ * q₃| / r₂₃²
Since we want F₁₃ + F₂₃ = 0, we can write:
k * |q₁ * q₃| / r₁₃² + k * |q₂ * q₃| / r₂₃² = 0
Let's plug in the given values:
q₁ = 1.00 μC = 1.00 x 10^-6 C (positive charge at the origin)
q₂ = -2.00 μC = -2.00 x 10^-6 C (negative charge at x = 10.0 cm = 0.10 m)
Now, we can calculate the distances r₁₃ and r₂₃ in terms of x.
1. Distance between q₁ and q₃ (r₁₃):
r₁₃ = |x - 0|
2. Distance between q₂ and q₃ (r₂₃):
r₂₃ = |x - 0.10|
Substituting these values into the equation above, we get:
k * |q₁ * q₃| / (x - 0)² + k * |q₂ * q₃| / (x - 0.10)² = 0
Simplifying further:
k * |q₁ * q₃| / x² + k * |q₂ * q₃| / (x - 0.10)² = 0
Now we can substitute the given values for q₁ and q₂:
k * |(1.00 x 10^-6) * q₃| / x² + k * |(-2.00 x 10^-6) * q₃| / (x - 0.10)² = 0
Dividing through by k and multiplying through by x²(x - 0.10)²:
|q₃| / x² + |-2.00 * q₃| / (x - 0.10)² = 0
Simplifying further:
|q₃| / x² - 2.00 * |q₃| / (x - 0.10)² = 0
Now, let's simplify the equation using the fact that |a|/b = a/b if a and b have the same sign, or -a/b if they have opposite signs:
q₃ / x² - 2.00 * q₃ / (x - 0.10)² = 0
Multiplying through by x²(x - 0.10)²:
q₃(x - 0.10)² - 2.00 * q₃x² = 0
Expanding the brackets:
q₃(x² - 0.20x + 0.01) - 2.00q₃x² = 0
Simplifying further:
q₃x² - 0.20q₃x + 0.01q₃ - 2.00q₃x² = 0
Combining like terms:
-1.00q₃x² - 0.20q₃x + 0.01q₃ = 0
Now, we have a quadratic equation in terms of x. We can solve this equation to find the position along the x-axis where a third charge experiences no force. However, note that we do not have the value of q₃. Once you provide the value of q₃, we can proceed with solving the equation and finding the solution for x.