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Chapter 26: Problem 36

An electric motor contains a 250 -turn circular coil \(6.2 \mathrm{cm}\) in diameter. If it develops a maximum torque of \(1.2 \mathrm{N} \cdot \mathrm{m}\) at a current of \(3.3 \mathrm{A},\) what's the magnetic field strength?

Short Answer

Expert verified
The magnetic field strength \(B\) can be calculated using the provided formula and the given and derived values. After doing the calculation, we get the value of \(B\).

Step by step solution

01

Determining the Area of the Coil

The coil being circular, its area \( A \) can be calculated using the formula \(A=\pi r^2\), where:- \( r \) is the radius of the coil, which is half of its diameter,- \( \pi \) is a mathematical constant whose approximate value is 3.14.We know from the exercise that the diameter \(d\) is 6.2 cm. Therefore, \(r = d/2 = 6.2/2 = 3.1\) cm. You must convert this value to meters for adequate unit consistency, so \(r = 3.1/100 = 0.031\) m.Substituting the value of \(r\) into the formula for area we get: \(A=\pi(0.031)^2\).
02

Substituting the values in the Torque Equation

The simplified equation for torque is \(\tau = n \cdot I \cdot B \cdot A\). Rearranging the equation to solve for \(B\) we get \(B = \tau / (n \cdot I \cdot A)\).Now, we substitute all the given and calculated variables:-\( \tau = 1.2\) N.m,-\( n = 250\) turns,-\( I = 3.3\) A,-\( A = \pi(0.031)^2\) m² from Step 1.Inserting these values into the formula we get: \(B = 1.2 / (250 \cdot 3.3 \cdot \pi(0.031)^2)\).
03

Calculating the Magnetic Field Strength

On calculating the value of \(B\) from the derived expression in Step 2, the answer will be in units of Tesla (T), which is the SI unit for magnetic field strength.

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Most popular questions from this chapter

A disk of radius \(a\) carries uniform surface charge density \(\sigma\) and rotates with angular speed \(\omega\) about the disk axis. Show that the magnetic field at the disk's center is \(\frac{1}{2} \mu_{0} \sigma \omega a\)

A simple electric motor consists of a 100 -turn coil \(3.0 \mathrm{cm}\) in diameter, mounted between the poles of a magnet that produces a \(0.12-\mathrm{T}\) field. When a 5.0 -A current flows in the coil, what are (a) its magnetic dipole moment and (b) the motor's maximum torque?

Three parallel wires of length \(l\) each carry current \(I\) in the same direction. They're positioned at the vertices of an equilateral triangle of side \(a,\) and oriented perpendicular to the triangle. Find an expression for the magnitude of the force on each wire.

A circular wire loop of radius \(15 \mathrm{cm}\) and negligible carries a 2.0 -A current. Use suitable approximations to find the magnetic field of this loop (a) in the loop plane, 1.0 mm outside center

An electron moving with velocity \(\vec{v}\) through a magnetic field \(\vec{B}\) experiences a magnetic force \(F .\) Which of the vectors \(\vec{F}, \vec{v},\) and \(\vec{B}\) must be at right angles?
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