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Why magnetic field is zero outside a solenoid ?

The assertion that the magnetic field is nearly zero outside a solenoid is a simplification that holds true under certain idealized conditions. A solenoid is a coil of wire wound in a helical shape, often used to generate a magnetic field. Let’s explore the reasons behind the statement that the magnetic field is zero or very weak outside a solenoid:

1. Idealized Model:

  • Assumption of Infinite Length: The statement that the magnetic field is zero outside a solenoid is often based on the assumption of an infinitely long solenoid. In an idealized model of an infinitely long solenoid, the magnetic field does become negligible outside the solenoid.
  • Cylindrical Symmetry: The symmetry of the solenoid, with the magnetic field lines forming concentric circles around the axis of the solenoid, contributes to the simplification of the external magnetic field.

2. Biot-Savart Law and Ampère’s Circuital Law:

  • Biot-Savart Law: The Biot-Savart Law describes the magnetic field produced by a current-carrying wire. When applied to a solenoid, the individual contributions of the current loops inside the solenoid add up, creating a significant magnetic field inside.
  • Ampère’s Circuital Law: Ampère’s Circuital Law relates the magnetic field along a closed loop to the current passing through the loop. In the case of a solenoid, the loop is chosen to encircle the solenoid’s axis, leading to a simplified expression for the magnetic field inside.

3. Cancellation of Magnetic Fields:

  • Equal and Opposite Contributions: Inside the solenoid, the magnetic fields produced by individual current loops reinforce each other, creating a strong and uniform magnetic field along the axis of the solenoid.
  • Cancellation Outside: As you move away from the axis of the solenoid, the contributions from different current loops start to cancel each other out. The magnetic fields produced by adjacent turns of the coil tend to have equal and opposite components, leading to a net cancellation.

4. Finite Length Considerations:

  • Field Leakage: In reality, finite-length solenoids may exhibit some magnetic field leakage outside the coil due to imperfections, end effects, or non-uniform winding. However, this external field is often significantly weaker than the field inside the solenoid.
  • Fringing Fields: Near the ends of a finite solenoid, the magnetic field lines may curve outward, creating fringing fields that extend beyond the coil. These fringing fields contribute to the external magnetic field but are still weaker compared to the field along the axis.

5. Application of Gauss’s Law for Magnetism:

  • Gauss’s Law for Magnetism: In the context of magnetostatics, Gauss’s Law for Magnetism states that the net magnetic flux through any closed surface is zero. This implies that the magnetic field lines are continuous and do not have sources or sinks.
  • Closed Surface Outside the Solenoid: If a closed surface is chosen outside the solenoid, the net magnetic flux through this surface must be zero according to Gauss’s Law. This condition is consistent with the idea that the magnetic field is negligible outside the solenoid.

6. Experimental Observations:

  • Experimental Verification: Experimental observations and measurements support the idea that the magnetic field is significantly weaker outside a solenoid compared to its interior. This observation aligns with the theoretical expectations based on the idealized model of an infinitely long solenoid.

In summary, the statement that the magnetic field is zero or very weak outside a solenoid is a simplification that arises from the idealized model of an infinitely long solenoid and the cancellation of magnetic fields due to the symmetry and arrangement of current loops within the solenoid. While external magnetic fields may exist in practice, they are often much weaker compared to the strong and uniform magnetic field present inside the solenoid.

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