To tackle this problem, we need to analyze the forces acting on the electron as it moves in the electric and magnetic fields created by the coaxial conducting pipes. The inner pipe, having a lower potential, creates an electric field that will influence the motion of the electron. Meanwhile, the uniform magnetic field will exert a magnetic force on the electron as it moves. Our goal is to find the conditions under which the electron cannot reach the outer pipe.

Understanding the Forces at Play

When the electron leaves the surface of the inner pipe, it experiences two main forces:

• Electric Force (F_E): This force is due to the electric field (E) created by the potential difference (V) between the two pipes. The electric field can be expressed as:

E = V / d, where d is the distance between the two pipes (b - a).

• Magnetic Force (F_B): As the electron moves through the magnetic field (B), it experiences a magnetic force given by:

F_B = q(v × B), where q is the charge of the electron, v is its velocity, and B is the magnetic field strength.

Analyzing the Motion of the Electron

Initially, the electron has negligible velocity as it leaves the inner pipe. The electric force will accelerate the electron towards the outer pipe. However, as it gains velocity, the magnetic force will also come into play. The direction of the magnetic force will be perpendicular to both the velocity of the electron and the magnetic field, which can cause the electron to spiral or deviate from a straight path.

Condition for Not Reaching the Outer Pipe

For the electron to not reach the outer pipe, the magnetic force must be strong enough to counteract the electric force effectively. This can happen if the magnetic field is sufficiently large. The condition can be expressed mathematically:

F_B ≥ F_E

Substituting the expressions for the forces, we have:

q(v × B) ≥ qE

Since the charge (q) of the electron is constant and can be canceled out, we simplify this to:

v × B ≥ E

Finding the Critical Magnetic Field Strength

Now, we need to express the electric field in terms of the potential difference and the distance:

E = V / (b - a)

For the electron to not reach the outer pipe, we need to ensure that the magnetic field strength B is such that:

v × B ≥ V / (b - a)

As the electron accelerates due to the electric field, its velocity v will increase. However, if we want to find the critical value of B, we can consider the scenario where the electron just barely reaches the outer pipe. This means we need to find the maximum velocity it can achieve under the influence of the electric field:

Using energy conservation, the kinetic energy gained by the electron as it moves from the inner to the outer pipe can be equated to the work done by the electric field:

(1/2)mv² = qV

From this, we can express the velocity:

v = √(2qV/m)

Final Expression for Magnetic Field

Substituting this expression for v back into our inequality gives:

√(2qV/m) × B ≥ V / (b - a)

Rearranging this leads us to the critical value of B:

B ≥ (V / (b - a)) × (m / √(2qV))

This inequality provides the threshold magnetic field strength above which the electron will not reach the outer pipe. If the magnetic field strength exceeds this value, the magnetic force will dominate, preventing the electron from reaching the outer pipe.

Conclusion

In summary, the interplay between the electric and magnetic forces determines the motion of the electron. By analyzing the forces and applying the principles of energy conservation, we can derive the conditions under which the electron cannot reach the outer pipe. This understanding is crucial in fields such as electromagnetism and particle physics, where charged particles interact with electric and magnetic fields.