A toroidal solenoid has a central radius of 0.50 m and a cross-sectional diameter of 10 cm. When a current passes through the coil of the solenoid, the magnetic field inside the solenoid at its CENTER has a magnitude of 2.0 µT. What is the largest value of the magnetic field inside the solenoid when this current is flowing? ( \mu_0 = 4\pi \times 10^{-7} \text{Tm/A}) Note: Your answer is assumed to be reduced to the highest power possible. Your Answer: x10 Answer Hide hint for Question 2 The answer should be with two decimals and without units.
A toroidal solenoid has a central radius of 0.50 m and a cross-sectional diameter of 10 cm. When a current passes through the coil of the solenoid, the magnetic field inside the solenoid at its CENTER has a magnitude of 2.0 µT. What is the largest value of the magnetic field inside the solenoid when this current is flowing? ( \mu_0 = 4\pi \times 10^{-7} \text{Tm/A}) Note: Your answer is assumed to be reduced to the highest power possible. Your Answer: x10 Answer Hide hint for Question 2 The answer should be with two decimals and without units.
The Correct Answer and Explanation is:
Correct Answer:
2.22 × 10⁻⁶
Explanation
A toroidal solenoid is a solenoid shaped like a doughnut, where the magnetic field is confined within the core. The magnetic field at a point inside the toroid is given by: B=μ0NI2πrB = \frac{\mu_0 N I}{2\pi r}
Where:
- BB is the magnetic field at a radius rr from the center,
- μ0=4π×10−7 T\cdotpm/A\mu_0 = 4\pi \times 10^{-7} \ \text{T·m/A},
- NN is the total number of turns,
- II is the current,
- rr is the radial distance from the center of the toroid.
The magnetic field varies inversely with the radius, so the field is strongest at the inner edge and weakest at the outer edge of the toroid.
Given Data:
- Central radius of the toroid: rc=0.50 mr_c = 0.50 \ \text{m}
- Cross-sectional diameter: d=0.10 m⇒d = 0.10 \ \text{m} \Rightarrow radius = 0.05 m0.05 \ \text{m}
- Magnetic field at center: Bcenter=2.0 μT=2.0×10−6 TB_{\text{center}} = 2.0 \ \mu\text{T} = 2.0 \times 10^{-6} \ \text{T}
Find the Largest Magnetic Field:
The maximum magnetic field occurs at the smallest radius inside the toroid: rmin=rc−d2=0.50−0.05=0.45 mr_{\text{min}} = r_c – \frac{d}{2} = 0.50 – 0.05 = 0.45 \ \text{m}
The field at the center is at r=0.50 mr = 0.50 \ \text{m}. Because B∝1rB \propto \frac{1}{r}, the field at r=0.45 mr = 0.45 \ \text{m} is: Bmax=Bcenter⋅rcrmin=2.0×10−6⋅0.500.45B_{\text{max}} = B_{\text{center}} \cdot \frac{r_c}{r_{\text{min}}} = 2.0 \times 10^{-6} \cdot \frac{0.50}{0.45} Bmax=2.0×10−6⋅1.1‾=2.22×10−6 TB_{\text{max}} = 2.0 \times 10^{-6} \cdot 1.\overline{1} = 2.22 \times 10^{-6} \ \text{T}
Final Answer:
2.22 × 10⁻⁶ (Expressed as 2.22 in the answer box with x10⁻⁶ understood)
