. Four point charges are separated by 1m in air as shown. If q1= 3nC,
q2= -4nC, q3= 2nC, and q4= -5nC, what is the electric field at a point halfway
between q2 and q3?
hi
To find the electric field at a point halfway between q2 and q3, we can calculate the electric field created by each charge individually at that point and then sum them up.
Step 1: Calculate the electric field created by each charge at the halfway point.
The electric field created by a point charge can be calculated using Coulomb's law:
Electric field (E) = (k * q) / r^2
Where:
- k is the electrostatic constant, approximately equal to 9 x 10^9 Nm^2/C^2
- q is the charge of the point charge
- r is the distance between the point charge and the point where we want to find the electric field
Let's calculate the electric field created by each charge at the halfway point:
Electric field created by q2 at the halfway point (E2):
E2 = (k * q2) / r2^2
Electric field created by q3 at the halfway point (E3):
E3 = (k * q3) / r3^2
Step 2: Determine the distances between each charge and the halfway point.
Given that the four point charges are separated by 1m, the distances between the halfway point and each charge can be calculated as follows:
- Distance between the halfway point and q2 (r2): 0.5m
- Distance between the halfway point and q3 (r3): 0.5m
Step 3: Calculate the electric fields created by each charge.
Using the values provided:
- q2 = -4nC
- q3 = 2nC
And the distance values calculated in Step 2:
- r2 = 0.5m
- r3 = 0.5m
Electric field created by q2 at the halfway point (E2):
E2 = (9 * 10^9 Nm^2/C^2 * -4nC) / (0.5m)^2
Electric field created by q3 at the halfway point (E3):
E3 = (9 * 10^9 Nm^2/C^2 * 2nC) / (0.5m)^2
Step 4: Calculate the total electric field at the halfway point.
To calculate the total electric field at the halfway point, we need to consider the signs of the charges. As q2 is negative and q3 is positive, their electric fields will have opposite directions. The total electric field is the vector sum of the individual electric fields:
Electric field at the halfway point (Et) = E2 - E3
Substituting the calculated values, we have:
Et = [ (9 * 10^9 Nm^2/C^2 * -4nC) / (0.5m)^2 ] - [ (9 * 10^9 Nm^2/C^2 * 2nC) / (0.5m)^2 ]
Et = [ (9 * -4 * 10^9 Nm^2/C^2 * nC) / (0.25m^2) ] - [ (9 * 2 * 10^9 Nm^2/C^2 * nC) / (0.25m^2) ]
Et = [ -36n / 0.25 ] - [ 18n / 0.25 ]
Et = -36n / 0.25 - 18n / 0.25
Et = - (36n + 18n) / 0.25
Et = - 54n / 0.25
Et = - 216n / 1
Et = - 216n
Therefore, the electric field at a point halfway between q2 and q3 is -216n C/N, pointing towards q2.
To find the electric field at a point halfway between q2 and q3, we can use the principle of superposition. This principle states that the electric field at a point due to multiple charges is the vector sum of the electric fields produced by each individual charge.
First, we need to calculate the electric field produced by each individual charge at the halfway point between q2 and q3. Let's call this point P.
We can use Coulomb's Law to calculate the electric field produced by each charge. The electric field produced by a point charge is given by the equation:
E = k * (Q / r^2)
Where:
E is the electric field,
k is the electrostatic constant (k = 9 x 10^9 Nm^2/C^2),
Q is the magnitude of the charge, and
r is the distance from the charge to the point.
Let's calculate the electric field produced by each charge.
For q2:
Q = -4nC
r = 0.5m (since the point is halfway between q2 and q3)
E2 = k * (Q2 / r^2)
E2 = 9 x 10^9 * (-4 * 10^-9 C) / (0.5)^2
E2 = -2.88 x 10^9 N/C
For q3:
Q = 2nC
r = 0.5m (since the point is halfway between q2 and q3)
E3 = k * (Q3 / r^2)
E3 = 9 x 10^9 * (2 * 10^-9 C) / (0.5)^2
E3 = 7.2 x 10^9 N/C
Now, we can calculate the net electric field at point P by adding the individual electric fields:
E_net = E2 + E3
E_net = -2.88 x 10^9 N/C + 7.2 x 10^9 N/C
E_net = 4.32 x 10^9 N/C
Therefore, the electric field at a point halfway between q2 and q3 is 4.32 x 10^9 N/C.