To determine the values of the potential difference (p.d.) V at which an electron can’t reach the outer pipe (anode) in a coaxial conductor setup, we need to analyze the forces acting on the electron as it moves from the inner pipe (cathode) to the outer pipe (anode). The setup involves a long straight wire carrying a current I along the axis of two coaxial conducting pipes with radii a (inner pipe) and b (outer pipe), where b > a. The electron starts from the surface of the inner pipe with negligible initial velocity.
Understanding the Electric Field
When a potential difference V is applied between the two pipes, an electric field is established in the region between them. The electric field (E) in a coaxial cylindrical conductor can be derived from Gauss's law. For a cylindrical shell, the electric field at a distance r from the axis (where a < r < b) is given by:
E(r) = (1 / (2πε₀)) * (I / (2πr))
Here, ε₀ is the permittivity of free space. This equation indicates that the electric field decreases with increasing distance from the axis of the wire.
Force on the Electron
The force (F) acting on the electron due to the electric field can be expressed as:
F = eE
where e is the charge of the electron (approximately 1.6 x 10-19 C). The direction of this force will be towards the outer pipe since the electron is negatively charged and the electric field points from the positive to the negative potential.
Energy Considerations
As the electron moves from the inner pipe to the outer pipe, it gains kinetic energy from the electric potential energy. The potential energy change (ΔU) as the electron moves through the potential difference V is given by:
ΔU = eV
For the electron to reach the outer pipe, it must have enough kinetic energy to overcome the electric field. If the electron starts with negligible velocity, its initial kinetic energy is essentially zero. Therefore, the work done on the electron must equal the potential energy gained:
KE = ΔU
Critical Potential Difference
To find the critical potential difference V at which the electron can no longer reach the outer pipe, we need to consider the maximum potential energy the electron can gain before it is repelled back. The condition for the electron to just reach the outer pipe can be expressed as:
eV = 0.5mv2
where m is the mass of the electron and v is the final velocity it needs to reach the outer pipe. If we set v to a very small value (approaching zero), we can see that the electron will not have enough energy to overcome the electric field if V is too low.
Final Expression for V
Thus, the critical potential difference V can be expressed in terms of the electric field and the distance the electron must travel:
V = (1 / e) * (0.5mv2)
However, since we are looking for the threshold where the electron cannot reach the outer pipe, we can conclude that:
- If V is less than a certain threshold value determined by the electric field strength and the distance between the pipes, the electron will not have enough energy to reach the outer pipe.
- As a result, the potential difference V must be sufficiently high to ensure that the electric field can impart enough energy to the electron to overcome any opposing forces.
In summary, the exact value of V will depend on the specific parameters of the system, including the current I, the radii of the pipes, and the distance the electron must travel. If V is too low, the electron will not be able to reach the outer pipe, effectively being trapped within the electric field created by the coaxial conductors.
