Electric charge charge is distributed uniformly along a thin rod of length "a", with total charge Q. Take the potential to be zero at infinity find the potential at P. If z>>a, what would be the potential at p? From potential find the electric field at point p for z>>a.
Note:
P is located above the middle of a, a distance of z
P
(distance vertical z)
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[-------a---------]
To find the potential at point P, we can use the principle of superposition. We'll consider small segments of the rod and integrate their contributions to obtain the total potential.
Since the charge is uniformly distributed, we can divide the rod into infinitesimally small segments Δz. Each segment will have a charge ΔQ = (Q/a)Δz.
The potential due to a small segment ΔV at P is given by the formula:
ΔV = k * (ΔQ)/r,
where k is the electrostatic constant (k = 1/4πε₀) and r is the distance from the segment to point P.
For z >> a, we can assume that the distance r is approximately equal to z (the distance from the rod to P).
Now, let's integrate the potential of each segment along the entire length of the rod:
V = ∫ ΔV = ∫[k * (ΔQ)/r]
The limits of integration will be from -a/2 to a/2, as these are the bounds of the rod.
V = ∫[-a/2 to a/2] [k * (ΔQ)/z]
Since the charge is uniformly distributed, we can express ΔQ = (Q/a)Δz. Substituting this into the equation:
V = kQ/a * ∫[-a/2 to a/2] [Δz/z]
To perform the integral, we'll use the natural logarithm:
V = kQ/a * ln(z)|[-a/2 to a/2]
V = kQ/a * [ ln(z) - ln(-a/2) + ln(a/2) ]
Now, we can use ln(-a/2) ≈ ln(a/2) to simplify the equation:
V = kQ/a * ln(z/a)
Since we took the potential to be zero at infinity, we can express the potential at point P as:
V = kQ/a * ln(z/a)
For z>>a, we can approximate ln(z/a) as ln(z), as compared to ln(a/2), ln(z) will dominate. Therefore, the potential at point P for z>>a is approximately:
V ≈ kQ/a * ln(z)
To find the electric field at point P, we can differentiate the potential with respect to z:
E = -dV/dz
E = -kQ/a * d(ln(z))/dz
As ln(z) differentiates to 1/z:
E = -kQ/(az)
So, the electric field at point P, for z>>a, is approximately:
E ≈ -kQ/(az)
To find the potential at point P, we can consider the electric field contribution from each differential segment of the rod and integrate them over the entire length of the rod.
1. The electric potential due to a point charge Q at a distance r is given by:
V = kQ / r
where k is the Coulomb's constant.
2. Let's assume the rod lies along the x-axis, and point P is at a distance z along the z-axis.
3. Consider a small segment dx of the rod at a distance x from the origin. The charge on this small segment can be considered as dq = Q*dx/a, as the charge is uniformly distributed over the entire length.
4. The distance between the small segment dx and point P can be calculated as r = sqrt(x^2 + z^2).
5. The potential due to the small charge segment dq at point P is given by:
dV = k*dq / r
= k*(Q*dx/a) / sqrt(x^2 + z^2)
6. To find the potential at P, we need to integrate the contributions from all the small charge segments of the rod:
V = ∫(k*(Q*dx/a)) / sqrt(x^2 + z^2) (from x = -a/2 to x = a/2)
Now let's consider the case when z >> a. In this limit, the distance z is much larger compared to the length of the rod.
In such a scenario, we can make an approximation by considering z as the dominant term in the denominator under the square root.
7. For z >> a, we can approximate the denominator sqrt(x^2 + z^2) ≈ z.
8. Integrating the potential expression with this approximation:
V ≈ ∫(k*(Q*dx/a) / z) (from x = -a/2 to x = a/2)
= (k*Q/a) * ∫dx / z (from x = -a/2 to x = a/2)
= (k*Q/a) * [x / z] (from x = -a/2 to x = a/2)
= (k*Q/a) * [a/2z - (-a/2z)]
= (k*Q/a) * (a/z)
= (k*Q) / z
So, when z >> a, the potential at point P becomes V = (k*Q) / z.
To find the electric field at point P, we can differentiate the potential with respect to z:
E = -dV / dz
= -(d/dz) [(k*Q) / z]
= -(k*Q) / z^2
Therefore, when z >> a, the electric field at point P is E = -(k*Q) / z^2.