Calculate the electric field at a point 3.12 cm perpendicular to the midpoint of a 2.23 m long thin wire carrying a total charge of 5.45 uC.
To calculate the electric field at a point due to a wire, you can use Coulomb's Law. Coulomb's Law states that the electric field (E) at a point due to a charge (q) is equal to the force between the charge and the point (F) divided by the distance (r) between them.
In this case, we have a wire carrying a total charge (Q), and we want to find the electric field at a point perpendicular to the midpoint of the wire. We can assume that the wire is infinitely long for simplicity.
To calculate the electric field, we need to use the concept of electric field due to a line charge. The electric field due to a small segment of the wire is given by:
dE = (k * dq) / r^2
where dE is the electric field due to a segment of the wire, k is the electrostatic constant (8.99 x 10^9 Nm^2/C^2), dq is the charge element of the wire, and r is the distance between the wire segment and the point where we want to calculate the electric field.
To find the electric field at the specified point, we need to integrate this equation over the entire length of the wire. However, since the wire is thin, we can approximate it as a point charge at the center of the wire.
The charge per unit length (λ) of the wire is given by:
λ = Q / L
where λ is the charge per unit length, Q is the total charge, and L is the length of the wire.
Now, we can substitute the equation for λ into the equation for dE:
dE = (k * λ * dx) / r^2
where dx is the differential length of the wire segment.
To integrate this equation over the entire length of the wire, we need to express dx in terms of the length variable. Since the wire is thin, we can approximate the length of the wire segment as dx ≈ dy.
Now, we can integrate dE from -L/2 to L/2 to find the electric field at the point:
E = ∫[(-L/2)^L/2] (k * λ * dy) / r^2
E = ∫[(-L/2)^L/2] (k * Q / L * dy) / r^2
E = (k * Q / L) * ∫[(-L/2)^L/2] (dy) / r^2
Simplifying the integral:
E = (k * Q / L) * (y/r^2) |_(y=-L/2)^(y=L/2)
E = (k * Q / L) * [((L/2) / r^2) - ((-L/2) / r^2)]
E = (k * Q / (L * r^2)) * (L/2 + L/2)
Finally, we can simplify the expression:
E = (k * Q) / (2 * L * r^2)
Plugging in the given values:
E = (8.99 x 10^9 Nm^2/C^2 * 5.45 x 10^-6 C) / (2 * 2.23 m * (0.0312 m)^2)
E ≈ 3.33 x 10^5 N/C
Therefore, the electric field at a point 3.12 cm perpendicular to the midpoint of a 2.23 m long thin wire carrying a total charge of 5.45 μC is approximately 3.33 x 10^5 N/C.