A +5.00 C charge is located at the origin. A-3.00C charge is placed at x = 1.00 m. At what finite distance(s) along the x-axis will the electric field be equal to zero?
in simple steps
To find the distance along the x-axis where the electric field is equal to zero, we can use the equation for electric field due to a point charge:
E = k * Q / r^2
where E is the electric field, k is the Coulomb's constant (9.0 x 10^9 Nm^2/C^2), Q is the charge, and r is the distance from the charge.
Given:
Q1 = +5.00 C (charge at the origin)
Q2 = -3.00 C (charge at x = 1.00 m)
To find the distance where the electric field is zero, we need to find the point on the x-axis where the electric field due to Q1 and Q2 cancel out.
Let's consider the electric field due to Q1 at a distance x from the origin:
E1 = k * Q1 / x^2
Similarly, let's consider the electric field due to Q2 at a distance (1.00 - x) from the origin:
E2 = k * Q2 / (1.00 - x)^2
For the total electric field to be zero, the magnitudes of E1 and E2 must be equal. Therefore, we can set up the equation:
E1 = E2
k * Q1 / x^2 = k * Q2 / (1.00 - x)^2
Canceling out the k in both sides of the equation:
Q1 / x^2 = Q2 / (1.00 - x)^2
Now, we can substitute the values:
+5.00 / x^2 = -3.00 / (1.00 - x)^2
Cross-multiplying:
+5.00 * (1.00 - x)^2 = -3.00 * x^2
Expanding and rearranging the equation:
5.00 - 10.00x + 5.00x^2 = -3.00x^2
8.00x^2 - 10.00x + 5.00 = 0
This equation is a quadratic equation. To solve it, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
where a = 8.00, b = -10.00, and c = 5.00
Plugging in these values:
x = (-(-10.00) ± √((-10.00)^2 - 4(8.00)(5.00))) / (2(8.00))
x = (10.00 ± √(100.00 - 160.00)) / 16.00
x = (10.00 ± √(-60.00)) / 16.00
Since the square root of a negative number is not a real number, this equation has no real solutions. Therefore, there are no finite distances along the x-axis where the electric field is zero in this scenario.
Sure! Here are the steps to find the finite distance(s) along the x-axis where the electric field is equal to zero:
Step 1: Determine the equation for the electric field due to each charge. The electric field due to a point charge can be calculated using the equation:
E = k * q / r^2
Where E is the electric field, k is the electrostatic constant (8.99 × 10^9 Nm^2/C^2), q is the charge, and r is the distance between the charges.
Step 2: Find the electric field for each charge separately. Let's calculate the electric field at a generic point (x, 0) on the x-axis due to the +5.00C charge located at the origin:
E1 = k * q1 / r1^2
= (8.99 × 10^9 Nm^2/C^2) * (+5.00 C) / (x^2)
Step 3: Calculate the electric field at the same generic point (x, 0) on the x-axis due to the -3.00C charge located at x = 1.00 m:
E2 = k * q2 / r2^2
= (8.99 × 10^9 Nm^2/C^2) * (-3.00 C) / ((x - 1)^2)
Step 4: Add the electric fields due to each charge to find the total electric field:
E_total = E1 + E2
Step 5: Set the total electric field (E_total) equal to zero and solve for x:
0 = E_total
0 = (8.99 × 10^9 Nm^2/C^2) * (+5.00 C) / (x^2) + (8.99 × 10^9 Nm^2/C^2) * (-3.00 C) / ((x - 1)^2)
Step 6: Simplify the equation and solve for x. This may involve algebraic manipulations, such as multiplying through by common denominators and moving terms around. The resulting equation will be a quadratic equation.
Step 7: Solve the quadratic equation to find the values of x where the electric field is zero. This can be done by factoring, using the quadratic formula, or by completing the square.
Once you solve the quadratic equation, you will find the finite distance(s) along the x-axis where the electric field is equal to zero.
To find the distance(s) along the x-axis where the electric field is zero, you can use the principle of superposition and apply Coulomb's law.
Step 1: Set up the problem
First, it's important to note that the electric field at any point along the x-axis due to multiple charges can be obtained by summing the electric fields due to each charge individually.
Step 2: Calculate the electric field due to the +5.00 C charge
The electric field, E1, at a point from a point charge is given by Coulomb's law:
E1 = k * q1 / r1^2
where k is the electrostatic constant (k = 9.0 × 10^9 N*m^2/C^2), q1 is the charge (5.00 C), and r1 is the distance from the origin (which is 0 in this case since the charge is located at the origin).
Since the positive charge contributes to a positive electric field along the x-axis, E1 will be positive.
Step 3: Calculate the electric field due to the -3.00 C charge
Similarly, the electric field, E2, at a point from a point charge is given by Coulomb's law:
E2 = k * q2 / r2^2
where q2 is the charge (-3.00 C), and r2 is the distance from the point charge located at x = 1.00 m.
Since the negative charge contributes to a negative electric field along the x-axis, E2 will be negative.
Step 4: Set up the equation for a zero electric field
To find where the electric field is zero, the sum of the electric fields from both charges needs to be equal to zero:
E1 + E2 = 0
Step 5: Solve the equation for the distance(s)
Substituting the values of E1 and E2, the equation becomes:
k * q1 / r1^2 - k * q2 / r2^2 = 0
Solve this equation for the distance(s) r2, which will give you the point(s) along the x-axis where the electric field is zero.
Step 6: Evaluate the final equation
Plug in the known values: q1 = +5.00 C, q2 = -3.00 C, r1 = 0, and solve for r2.