Two point charges 5uc and 3uc are fixed 4cm apart . calculate the distance in between them at which the resultant is zero
show calculation
5/x^2 = 3/y^2
x+y= 4 so y = (4-x)
5/x^2 = 3 / (4-x)^2
5(16 - 8 x + x^2) = 3 x^2
80 - 40 x + 5 x^2 = 3 x^2
2 x^2 - 40 x + 80 = 0
x^2 - 20 x + 40 = 0
realistic root at 2.254
check
x = 2.254
y = 4-2.254 = 1.746
5/x^2 = .984
3/y^2 = .984
yeah, that works
To find the distance at which the resultant electric field is zero between two point charges, we can use the principle of superposition. The resultant electric field is zero when the electric fields created by each charge cancel each other out.
Let's calculate the distance between the two charges at which the resultant electric field is zero.
Given:
Charge q1 = 5 μC
Charge q2 = 3 μC
Separation distance r = 4 cm = 0.04 m (converted to meters)
Electric constant k = 8.99 × 10^9 Nm²/C²
The electric field created by each charge is given by the formula:
E = (k * q) / (r^2)
The total electric field at the zero resultant point is the vector sum of the electric fields due to each charge. Therefore, the absolute values of the two electric fields should be equal, but opposite in direction.
Setting up the equation:
(k * q1) / (r₁^2) = -(k * q2) / (r₂^2)
We need to find the distance r at which this equation is satisfied.
Substituting the given values:
(8.99 × 10^9) * (5 × 10^-6) / (r₁^2) = -((8.99 × 10^9) * (3 × 10^-6)) / (r₂^2)
Simplifying the equation:
r₂^2 = (3/5) * (r₁^2)
Taking the square root of both sides:
r₂ = sqrt((3/5) * (r₁^2))
Now, substituting the given values:
r₂ = sqrt((3/5) * (0.04^2))
= sqrt((3/5) * 0.0016)
= sqrt(0.00096)
= 0.031 m or 3.1 cm
Therefore, the distance between the two charges at which the resultant electric field is zero is approximately 0.031 meters or 3.1 cm.
To calculate the distance between two point charges at which the resultant electric field is zero, we can use the concept of Coulomb's Law.
First, let's assign some values to the given information:
- Charge 1 (q1) = 5 µC (microCoulombs)
- Charge 2 (q2) = 3 µC
- Distance between the charges (r) = 4 cm = 0.04 m
Now, Coulomb's Law states that the electric force between two charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. In equation form:
Electric Force (F) = (k * q1 * q2) / r^2
Where:
F is the electric force,
k is the electrostatic constant (9 × 10^9 Nm^2/C^2),
q1 and q2 are the magnitudes of the charges, and
r is the distance between the charges.
To find the distance at which the resultant electric field is zero, we need to find a point on the line joining the two charges where the electric field due to each charge cancels out.
The electric field (E) produced by a point charge is given by:
E = F / q
Since the resultant electric field is zero at the point where the charges are located, we can set up the following equation:
E1 = E2
(F1 / q1) = (F2 / q2)
(k * q1 * q) / r^2 = (k * q2 * (r - q)) / r^2
Simplifying the equation:
q1 = q2 * (r - q)
Now, substitute the given values:
5 µC = 3 µC * (r - 0.04 m)
Solving this equation for r:
5 µC = 3 µC * r - 0.12 µC
0.12 µC = 3 µC * r - 5 µC
0.12 µC + 5 µC = 3 µC * r
5.12 µC = 3 µC * r
r = 5.12 µC / 3 µC
r ≈ 1.71 m
Therefore, the distance between the charges at which the resultant electric field is zero is approximately 1.71 meters.