The magnetic field outside an ideal solenoid is approximately zero because the net current enclosed by an Amperian loop outside the solenoid is zero, according to Ampère's Law.
Explanation:
The core principle explaining why the magnetic field is zero outside a solenoid lies in Ampère's Law. Ampère's Law relates the integrated magnetic field around a closed loop to the electric current passing through the loop. Mathematically, it's expressed as:
∮ B ⋅ dl = μ₀I_enc
Where:
- B is the magnetic field
- dl is an infinitesimal length element of the closed loop
- μ₀ is the permeability of free space
- I_enc is the net current enclosed by the loop
Applying Ampère's Law Outside a Solenoid:
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Consider an Amperian Loop: Imagine drawing a closed loop outside the solenoid that encircles the entire solenoid.
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Currents Enclosed: The solenoid consists of many turns of wire carrying current. These turns effectively create currents flowing in opposite directions through the loop we've imagined.
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Net Current: The current flows up on one side of the solenoid and down on the other side. Because the loop encircles the entire solenoid, the total current going into the loop is equal to the total current going out of the loop. Therefore, the net current (I_enc) enclosed by the loop is zero.
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Zero Magnetic Field: According to Ampère's Law, if the net current is zero, the magnetic field must be zero (Reference: Provided document dated 09-Jul-2024). Since I_enc = 0, then ∮ B ⋅ dl = 0. This implies that the magnetic field (B) outside the solenoid is zero.
Summary:
| Aspect | Description |
|---|---|
| Principle | Ampère's Law |
| Amperian Loop | Drawn outside the solenoid, enclosing the entire structure. |
| Net Current | Zero, because the currents flowing in opposite directions cancel each other out. |
| Magnetic Field | Zero, as a direct consequence of Ampère's Law and the zero net current. |
