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Question: The magnetic field inside a 200 turns solenoid of radius 10 cm is 2.9 × 10 − 4 T e s l a . If the so...

The magnetic field inside a 200 turns solenoid of radius 10 cm is 2.9 × 10 − 4 T e s l a . If the solenoid carries a current of 0.29 A, then the length of the solenoid is ________ π cm .

Answer

8

Explanation

Solution

The magnetic field (BB) inside a long solenoid is given by the formula: B=μ0nIB = \mu_0 n I where:

  • μ0\mu_0 is the permeability of free space, μ0=4π×10−7 T m/A\mu_0 = 4\pi \times 10^{-7} \text{ T m/A}.
  • nn is the number of turns per unit length.
  • II is the current flowing through the solenoid.

The number of turns per unit length (nn) can be expressed as the total number of turns (NN) divided by the length of the solenoid (LL): n=NLn = \frac{N}{L} Substituting this into the magnetic field formula, we get: B=μ0NLIB = \mu_0 \frac{N}{L} I

We are given:

  • Number of turns, N=200N = 200.
  • Radius of the solenoid, R=10 cm=0.1 mR = 10 \text{ cm} = 0.1 \text{ m} (Note: The radius is not needed for this calculation, as the formula for the magnetic field inside a long solenoid is independent of the radius).
  • Magnetic field, B=2.9×10−4 TB = 2.9 \times 10^{-4} \text{ T}.
  • Current, I=0.29 AI = 0.29 \text{ A}.

We need to find the length of the solenoid, LL. Rearranging the formula to solve for LL: L=μ0NIBL = \frac{\mu_0 N I}{B}

Now, substitute the given values into the equation: L=(4π×10−7 T m/A)×(200)×(0.29 A)2.9×10−4 TL = \frac{(4\pi \times 10^{-7} \text{ T m/A}) \times (200) \times (0.29 \text{ A})}{2.9 \times 10^{-4} \text{ T}}

Let's perform the calculation: L=4π×10−7×200×0.292.9×10−4 mL = \frac{4\pi \times 10^{-7} \times 200 \times 0.29}{2.9 \times 10^{-4}} \text{ m} We can simplify this expression: L=4π×200×0.292.9×10−710−4 mL = 4\pi \times \frac{200 \times 0.29}{2.9} \times \frac{10^{-7}}{10^{-4}} \text{ m} Notice that 0.292.9=0.1\frac{0.29}{2.9} = 0.1. L=4π×(200×0.1)×10−7−(−4) mL = 4\pi \times (200 \times 0.1) \times 10^{-7 - (-4)} \text{ m} L=4π×20×10−3 mL = 4\pi \times 20 \times 10^{-3} \text{ m} L=80π×10−3 mL = 80\pi \times 10^{-3} \text{ m}

The question asks for the length in units of π\pi cm. First, convert the length from meters to centimeters: 1 m=100 cm1 \text{ m} = 100 \text{ cm} L=80π×10−3 m×100 cm1 mL = 80\pi \times 10^{-3} \text{ m} \times \frac{100 \text{ cm}}{1 \text{ m}} L=80π×10−3×102 cmL = 80\pi \times 10^{-3} \times 10^2 \text{ cm} L=80π×10−1 cmL = 80\pi \times 10^{-1} \text{ cm} L=8π cmL = 8\pi \text{ cm}

The question states: "then the length of the solenoid is ________ π\pi cm". Comparing our result 8π8\pi cm with the blank format, the value to fill in the blank is 8.