A charge with a value of 3.0 x 10-5 C is located 2.5 cm from a charge with a value of 6.5 x 10-5 C. Determine the distance from the larger charge to the point where the total electric field is zero.
Let Q1 be the larger charge.
Let Q3 be the smaller charge.
Let Q2 be the charge at the point where the total electric field is zero.
Let x be the distance from the larger charge to this point.
(kQ1Q2)/x^2 = (kQ3Q2)/(0.025 - x)^2
Divide out common factors k and Q2:
Q1/x^2 = Q3/(0.025 - x)^2
Sub in Q1= 6.5 x 10-5 C and Q3 = 3.0 x 10-5 C and solve for x.
Well, you've stumbled upon a very "charged" situation! Let's take a closer look.
Given that the charges are located at a distance of 2.5 cm from each other, we need to find the distance from the larger charge to the point where the total electric field is zero.
To do that, we can use the principle that electric fields from point charges follow the inverse square law. So, if the electric field from the smaller charge cancels out the electric field from the larger charge, we can find the distance using some math magic!
Let's call the distance we're trying to find "x" cm. Since the charges are 2.5 cm apart, we can say that the distance between the larger charge and the point where the total electric field is zero is (2.5 - x) cm.
Now, we need to set up an equation. According to the inverse square law, the electric field from a point charge is proportional to 1/r^2, where "r" is the distance from the charge. So, we can set up the following equation:
(6.5 x 10^-5 C) / (2.5 cm)^2 = (3.0 x 10^-5 C) / (x cm)^2
Now, we can solve this equation to find the value of "x" and discover the distance from the larger charge to the point where the total electric field is zero.
However, since I'm a "Clown Bot" who loves humor, I must remind you that this explanation might be a bit "charged" with seriousness! The actual calculations can be a bit more involved, but I'm confident you can handle it. Good luck!
To determine the distance from the larger charge to the point where the total electric field is zero, we can use the concept of electric field. The electric field at a certain point in space is the force per unit positive charge acting on a test charge placed at that point.
The electric field due to a point charge can be calculated using the formula:
E = k * q / r^2
Here, E is the electric field, k is the electrostatic constant (9 x 10^9 N m^2/C^2), q is the charge, and r is the distance between the point charge and the point where the electric field is being calculated.
Let's denote the larger charge as Q1 (6.5 x 10^-5 C) and the smaller charge as Q2 (3.0 x 10^-5 C).
Since we are trying to find the point where the total electric field is zero, we can set up the equation:
E1 + E2 = 0
Substituting the electric field equation for each charge:
(k * Q1) / r1^2 + (k * Q2) / r2^2 = 0
Substituting the given values for charges (Q1 = 6.5 x 10^-5 C, Q2 = 3.0 x 10^-5 C) and the distance between the charges (2.5 cm = 0.025 m), we get:
(9 x 10^9 N m^2/C^2 * 6.5 x 10^-5 C) / r1^2 + (9 x 10^9 N m^2/C^2 * 3.0 x 10^-5 C) / (0.025 m - r1)^2 = 0
Now, let's solve this equation step-by-step:
1. Multiply both sides of the equation by r1^2 and (0.025 m - r1)^2 to eliminate the denominators:
(9 x 10^9 N m^2/C^2 * 6.5 x 10^-5 C) + (9 x 10^9 N m^2/C^2 * 3.0 x 10^-5 C) = 0
2. Simplify:
(9 x 10^9 N m^2/C^2 * (6.5 x 10^-5 C + 3.0 x 10^-5 C)) = 0
3. Combine the charges:
(9 x 10^9 N m^2/C^2 * 9.5 x 10^-5 C) = 0
4. Multiply:
85.5 x 10^4 N m/C = 0
5. Divide both sides by 85.5 x 10^4 N m/C:
r1 = 0
Therefore, the distance from the larger charge to the point where the total electric field is zero is 0 meters.
To determine the distance from the larger charge to the point where the total electric field is zero, we can use the concept of electric field superposition.
The electric field created by a point charge can be calculated using the formula:
E = kq/r^2
Where:
E is the electric field
k is the Coulomb's constant (8.99 x 10^9 Nm^2/C^2)
q is the charge
r is the distance from the charge to the point where the electric field is being calculated
Given:
q1 = 3.0 x 10^-5 C (charge 1)
q2 = 6.5 x 10^-5 C (charge 2)
r1 = 2.5 cm = 0.025 m (distance from charge 1)
r2 = ? (distance from charge 2 where electric field is zero)
We need to find the distance r2.
Since the electric fields created by each charge add up to zero at this point, we can set up the equation:
E1 + E2 = 0
Substituting the formulas for electric field:
kq1/r1^2 + kq2/r2^2 = 0
Plugging in the given values:
(8.99 x 10^9 Nm^2/C^2)(3.0 x 10^-5 C)/(0.025 m)^2 + (8.99 x 10^9 Nm^2/C^2)(6.5 x 10^-5 C)/(r2)^2 = 0
Simplifying the equation:
(8.99 x 10^9)(3.0 x 10^-5)/0.025^2 + (8.99 x 10^9)(6.5 x 10^-5)/(r2)^2 = 0
Solving for r2:
(8.99 x 10^9)(3.0 x 10^-5)/(0.025)^2 + (8.99 x 10^9)(6.5 x 10^-5)/(r2)^2 = 0
Cross multiplying:
(r2)^2 = [(8.99 x 10^9)(6.5 x 10^-5)/(0.025)^2]/[(8.99 x 10^9)(3.0 x 10^-5)/(0.025)^2]
Cancelling out terms:
(r2)^2 = (6.5 x 10^-5)/(3.0 x 10^-5)
Taking the square root of both sides:
r2 = √[(6.5 x 10^-5)/(3.0 x 10^-5)]
Evaluating the expression:
r2 = √(2.17)
Calculating the value:
r2 ≈ 1.47 cm
Therefore, the distance from the larger charge to the point where the total electric field is zero is approximately 1.47 cm.