A thin straight wire 20 cm in length carries a charge of one microcoulomb. Calculate the electric intensity and potential at a point P on the axis of the wire at a distance of 30 cm from the mid point of the wire.
To calculate the electric intensity at point P due to the wire, we can use the formula:
Electric Intensity (E) = k * (Q / r^2)
Where:
- k is the electrostatic constant (k = 9 × 10^9 Nm^2/C^2)
- Q is the charge on the wire (Q = 1 μC = 1 × 10^-6 C)
- r is the distance between the point P and the charge on the wire (r = 30 cm = 0.3 m)
Substituting the values into the formula, we get:
E = (9 × 10^9 Nm^2/C^2) * ((1 × 10^-6 C) / (0.3 m)^2)
E = (9 × 10^9 Nm^2/C^2) * (1 × 10^-6 C) / (0.09 m^2)
E = (9 × 10^9 / 0.09) N/C
E = 1 × 10^11 N/C
Therefore, the electric intensity at point P is 1 × 10^11 N/C.
To calculate the potential at point P due to the wire, we can use the formula:
Potential (V) = k * (Q / r)
Substituting the values into the formula, we get:
V = (9 × 10^9 Nm^2/C^2) * ((1 × 10^-6 C) / (0.3 m))
V = (9 × 10^9 Nm^2/C^2) * (1 × 10^-6 C) / (0.3 m)
V = (9 × 10^9 / 0.3) Nm/C
V = 3 × 10^10 Nm/C
Therefore, the potential at point P is 3 × 10^10 Nm/C.
To calculate the electric intensity and potential at point P on the axis of the wire, we can make use of the formula for the electric field due to a uniformly charged straight wire.
The electric intensity, or electric field strength (E), at a point P on the axis of the wire is given by the formula:
E = (k * λ) / r
where:
- E is the electric intensity in N/C (newtons per coulomb)
- k is the electrostatic constant, approximately equal to 9 × 10^9 Nm^2/C^2
- λ is the linear charge density in C/m (coulombs per meter)
- r is the distance of point P from the wire in meters
In this case, the wire has a length of 20 cm, so its total charge is 1 microcoulomb (1 μC). Therefore, the linear charge density λ is calculated as:
λ = Q / L
where:
- Q is the total charge on the wire in C (coulombs)
- L is the length of the wire in meters
Given that Q = 1 μC and L = 20 cm, we convert the units to Q = 1 × 10^(-6) C and L = 0.2 m.
Now, we can calculate the linear charge density:
λ = Q / L = (1 × 10^(-6) C) / (0.2 m) = 5 × 10^(-6) C/m
Next, we substitute the values into the formula for the electric intensity:
E = (k * λ) / r = (9 × 10^9 Nm^2/C^2) * (5 × 10^(-6) C/m) / (0.3 m)
Simplifying the expression gives:
E = 15 N/C
Therefore, the electric field intensity at point P is 15 N/C.
To calculate the potential at point P, we use the formula for the potential due to a point charge:
V = (k * Q) / r
where:
- V is the electric potential in volts (V)
- k is the electrostatic constant, approximately equal to 9 × 10^9 Nm^2/C^2
- Q is the total charge on the wire in C (coulombs)
- r is the distance of point P from the wire in meters
Substituting the values into the formula:
V = (9 × 10^9 Nm^2/C^2) * (1 × 10^(-6) C) / (0.3 m)
Simplifying the expression gives:
V = 30,000 V
Therefore, the electric potential at point P is 30,000 V.