Two fixed charges 1.07mC and -3.28mC are 61.8 cm apart. Where the third charge may be placed so that no net force acts on it
To find the location where the third charge may be placed so that no net force acts on it, we need to ensure that the net electrostatic force on the third charge is zero. This can be achieved by placing the third charge at a point where the electric fields due to the two fixed charges cancel out.
Let's call the two fixed charges Q1 and Q2, and the third charge Q3. The electrostatic force between two charges can be calculated using Coulomb's law:
F = k * (|Q1| * |Q2|) / r²
Where:
F is the force between the charges,
k is the electrostatic constant (k = 9.0 x 10^9 Nm²/C²),
|Q1| and |Q2| are the magnitudes of the charges, and
r is the distance between the charges.
To find the location where the net force is zero, we assume that Q3 is positive since Q1 and Q2 have opposite charges.
Let's proceed step-by-step:
Step 1: Calculate the forces between Q1 and Q3 and Q2 and Q3.
F1 = k * (Q1 * Q3) / r₁²
F2 = k * (Q2 * Q3) / r₂²
Where:
F1 is the force between Q1 and Q3,
F2 is the force between Q2 and Q3,
r₁ is the distance between Q1 and Q3, and
r₂ is the distance between Q2 and Q3.
Step 2: Set the forces equal to each other and solve for the distance r₁.
k * (Q1 * Q3) / r₁² = k * (Q2 * Q3) / r₂²
(Q1 * Q3) / r₁² = (Q2 * Q3) / r₂²
(Q1 / r₁²) = (Q2 / r₂²)
r₁² = (Q1 * r₂²) / Q2
r₁ = sqrt((Q1 * r₂²) / Q2)
Step 3: Substitute the given values and solve for r₁.
r₁ = sqrt((1.07mC * (0.618m)²) / 3.28mC)
r₁ = sqrt((1.07 * 10⁻³ C * (0.618 * 10⁻¹ m)²) / (3.28 * 10⁻³ C))
r₁ = sqrt(0.4127 * 10⁻⁵ m² / 3.28 * 10⁻³ C)
r₁ = sqrt(125.61 * 10⁻⁸ m² / 3.28 * 10⁻³ C)
r₁ = sqrt(3.835 * 10⁻⁵ m² / C)
r₁ ≈ 0.0062 m
Step 4: Now that we have the distance r₁, we can determine the position where the third charge can be placed. It is 0.0062 m away from Q1 along the line connecting Q1 and Q2.
Therefore, to achieve no net force on the third charge, it should be placed approximately 0.0062 m away from the positive charge and along the same line as Q1 and Q2.
To find the location where a third charge can be placed such that no net force acts on it, we need to consider the concept of electrostatic equilibrium. In electrostatic equilibrium, the net force on an object is zero.
We can use Coulomb's Law to calculate the force between the two fixed charges. Coulomb's Law states that the magnitude of the electrostatic force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.
The formula for Coulomb's Law is:
F = k * (|q1| * |q2|) / r^2
Where:
F is the magnitude of the electrostatic force,
k is Coulomb's constant (approximately equal to 9 × 10^9 N m^2/C^2),
q1 and q2 are the magnitudes of the charges, and
r is the distance between the charges.
In this case, we have two fixed charges, q1 = 1.07mC and q2 = -3.28mC, and the distance between them is r = 61.8 cm = 0.618 m.
Let's calculate the force (F) between the two charges:
F = (9 × 10^9 N m^2/C^2) * (|1.07mC| * |(-3.28mC)|) / (0.618m)^2
F = (9 × 10^9 N m^2/C^2) * (1.07 × 10^-6 C * 3.28 × 10^-6 C) / (0.618m)^2
F = (9 × 10^9 N m^2/C^2) * (3.5144 × 10^-12 C^2) / 0.381624m^2
F = 1.4126 × 10^-2 N
Now, to find the location where a third charge can be placed such that there is no net force acting on it, we need to consider the principle of superposition. This principle states that the net force on an object due to multiple charges is the vector sum of the individual forces exerted by each charge.
If the magnitude and direction of the forces exerted by the two fixed charges are equal and opposite, then their vector sum will be zero, ensuring no net force on the third charge.
Let's assume the third charge is q3, and it will be placed at a distance x from the charge q1 and at a distance (0.618 - x) from the charge q2.
The force exerted on q3 by charge q1 is given by:
F1 = k * (|q1| * |q3|) / x^2
The force exerted on q3 by charge q2 is given by:
F2 = k * (|q2| * |q3|) / (0.618 - x)^2
To ensure no net force, F1 and F2 should be equal and opposite. Therefore,
F1 = F2
k * (|q1| * |q3|) / x^2 = k * (|q2| * |q3|) / (0.618 - x)^2
Canceling out the k and |q3| on both sides:
(|q1| / x^2) = (|q2| / (0.618 - x)^2)
Now, we can substitute the given values of |q1|, |q2|, and solve for x:
(1.07 × 10^-6 C / x^2) = (3.28 × 10^-6 C / (0.618 - x)^2)
Cross-multiplying:
(1.07 × 10^-6 C) * ((0.618 - x)^2) = (3.28 × 10^-6 C) * (x^2)
Expanding ((0.618 - x)^2) on the left side:
(1.07 × 10^-6 C) * (0.618^2 - 2 * 0.618 * x + x^2) = (3.28 × 10^-6 C) * (x^2)
(1.07 × 10^-6 C) * (0.618^2 - 2 * 0.618 * x + x^2) = (3.28 × 10^-6 C) * (x^2)
1.07 × 10^-6 C * 0.618^2 - 2 * 1.07 × 10^-6 C * 0.618 * x + 1.07 × 10^-6 C * x^2 = 3.28 × 10^-6 C * x^2
Rearranging the equation:
(1.07 × 10^-6 C * 0.618^2) - (3.28 × 10^-6 C * x^2) = (2 * 1.07 × 10^-6 C * 0.618 * x)
Simplifying:
1.07 × 10^-6 C * (0.618^2 - 3.28 × 10^-6 C * x^2) = 1.325 × 10^-6 C * x
Dividing both sides by x:
1.07 × 10^-6 C * (0.618^2 - 3.28 × 10^-6 C * x^2) / x = 1.325 × 10^-6 C
Now, we need to solve this equation to find the value of x. This involves rearranging the equation to isolate and solve the quadratic term.