A 2.00 A current runs through a solenoid with 300 turns over 0.500 m. Calculate the magnetic field inside the solenoid. Also, calculate the magnetic field at a distance of 0.0400 m from a straight wire carrying the same current.

Question

BlackTom AI · Accepted Answer

We break down the problem into two parts: the magnetic field inside a solenoid and the magnetic field from a long straight wire.

First, solenoid: The magnetic field inside a long solenoid is B = μ0 n I, where n is the turns per unit length. Given N = 300 turns and length L = 0.500 m, n = N/L = 300 / 0.500 = 600 turns/m. With μ0 = 4π × 10^-7 T·m/A and current I = 2.00 A, we get B = (4π × 10^-7) × (600) × (2) = (4π × 10^-7) × 1200. Multiplying 4 × 1200 gives 4800, so B = 4800π × 10^-7 T ≈ 1.51 × 10^-3 T. This is the solenoid’s field.

Second, field from a long straight wire: The magnetic field at a distance r from a long straight wire carrying current I is B = μ0 I / (2π r). Here, I = 2.00 A and r = 0.0400 m. Substitute: B = (4π × 10^-7) × (2) / (2π × 0.0400). The π factors cancel, leaving B = (4 × 10^-7 × 2) / (2 × 0.0400) = (8 × 10^-7) / 0.08 = 1.0 × 10^-5 T. So the field at 0.0400 m from the wire is 1.0 × 10^-5 T.

Now compare to the provided options:
- Option a lists Solenoid = 1.51 × 10^-3 T (correct value for the solenoid) but Wire = 1.00 × 10^-4 T (too large by a factor of 10); this is inconsistent with the straight-wire formula. 
- Option b has Solenoid = 3.77 × 10^-3 T (which would require a larger n I than given, so this is incorrect) and Wire = 1.00 × 10^-5 T (the wire field matches the correct value). 
- Option c gives Solenoid = 1.51 × 10^-3 T and Wire = 1.00 × 10^-5 T (both values match the calculations above). 
- Option d lists Wire = 5.00 × 10^-6 T (which is half of the correct 1.00 × 10^-5 T), so this is incorrect.

In summary, the solenoid field calculation aligns with B = μ0 n I using n = 600 m^-1 and I = 2 A, giving 1.51 × 10^-3 T. The straight-wire field at 0.0400 m aligns with B = μ0 I / (2π r), giving 1.00 × 10^-5 T. The combination that matches both results is the pair Solenoid: 1.51 × 10^-3 T; Wire: 1.00 × 10^-5 T.

A 2.00 A current runs through a solenoid with 300 turns over 0.500 m. Calculate the magnetic field inside the solenoid. Also, calculate the magnetic field at a distance of 0.0400 m from a straight wire carrying the same current.单项选择题

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