1. Two point charges 4 μC and 2 μC are separated by a distance 1 m in air. At what point on the line joining the two charges is the electric field intensity zero?
let x be the distance from the 4microC charge
k4/x^2=k2/(1-x)^2
(1-x)^2=x^2 /2
1-2x+x^2=x^2 /2
x^2-4x+2=0
x=(4+-sqrt(16-8))/2= 2+-sqrt2
x=2+-1.41=
x=.59 m from the 4micro charge
check the math.
To find the point on the line joining the two charges where the electric field intensity is zero, we can use the principle of superposition. The electric field at any point due to these charges will be the vector sum of the electric fields produced by each charge individually.
Let's consider a point P on the line joining the two charges, and let's call the distance AP as x (distance from charge 4 μC) and PB as (1-x) (distance from charge 2 μC). According to the principle of superposition, the total electric field at point P is given by:
E_total = E_1 + E_2
where E_1 is the electric field due to the 4 μC charge at point P, and E_2 is the electric field due to the 2 μC charge at point P.
The electric field E_1 due to the 4 μC charge at point P is given by Coulomb's Law:
E_1 = k * (q_1) / r_1^2
where k is the electrostatic constant (9 × 10^9 Nm^2/C^2), q_1 is the charge of the 4 μC charge, and r_1 is the distance between the 4 μC charge and point P (which is x).
Similarly, the electric field E_2 due to the 2 μC charge at point P is given by:
E_2 = k * (q_2) / r_2^2
where q_2 is the charge of the 2 μC charge, and r_2 is the distance between the 2 μC charge and point P (which is 1-x).
Since we want to find the point on the line where the electric field is zero, we can set E_total equal to zero and solve for x.
E_total = E_1 + E_2 = 0
k * (q_1) / r_1^2 + k * (q_2) / r_2^2 = 0
Substituting the given values:
(9 × 10^9 Nm^2/C^2) * (4 μC) / x^2 + (9 × 10^9 Nm^2/C^2) * (2 μC) / (1-x)^2 = 0
Simplifying the equation and solving for x will give us the desired point on the line where the electric field intensity is zero.
To find the point on the line joining the two charges where the electric field intensity is zero, we need to apply the principle of superposition and consider the electric fields created by both charges. The electric field intensity due to a point charge at a given location can be calculated using Coulomb's law.
Coulomb's law states that the electric field intensity (E) created by a point charge (Q) at a distance (r) is given by the formula:
E = k * (Q / r^2)
Where:
- E is the electric field intensity
- Q is the charge
- r is the distance between the charge and the point where we want to calculate the electric field intensity
- k is the electrostatic constant, 8.99 x 10^9 Nm^2/C^2
For our case, we have two charges: +4 μC and +2 μC, separated by a distance of 1 m. We are looking for the point on the line joining these charges where the electric field intensity is zero.
Let's assume that the +4 μC charge is located at point A and the +2 μC charge is located at point B, with the line connecting them AB. We want to find a point P on the line AB where the electric field intensity is zero.
To find this point, we need to take into account the fact that the electric field created by the +4 μC charge at point P should be equal in magnitude and opposite in direction to the electric field created by the +2 μC charge.
Mathematically, we can express this as:
E1 = E2
Where:
- E1 is the electric field intensity at point P due to the +4 μC charge at point A
- E2 is the electric field intensity at point P due to the +2 μC charge at point B
Using Coulomb's law, we can calculate these electric field intensities separately:
E1 = k * (Q1 / r1^2)
E2 = k * (Q2 / r2^2)
Where:
- Q1 = +4 μC (charge at point A)
- Q2 = +2 μC (charge at point B)
- r1 = distance from point A to point P
- r2 = distance from point B to point P
Since we want the electric field intensities to be equal in magnitude and opposite in direction, we can set up the following equation:
k * (Q1 / r1^2) = -k * (Q2 / r2^2)
Simplifying the equation, we get:
Q1 / r1^2 = -Q2 / r2^2
Substituting the given values:
(4 μC) / r1^2 = -(2 μC) / r2^2
Now, we can solve this equation to find the values of r1 and r2, which will give us the distances from the charges to the point P where the electric field intensity is zero.
Once we have the distances r1 and r2, we can find the location of point P by measuring these distances from points A and B respectively.
Remember to always be cautious with the signs (+ and -) to ensure the correct direction of the electric fields.
kq1/r^2 = kq2/(1-r)^2
The k's will cross out, then cross multiply
q2 r^2 = q1 (1-r)^2
Then solve for r (you have a little math to do)