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Chapter 30: Q66P (page 900)

A circular loop of wire 50 mmin radius carries a current of 100 A. (a) Find the magnetic field strength. (b) Find the energy density at the center of the loop

Short Answer

Expert verified

$a) B = 1.3 脳 10^{- 3} T b) u_{B} = 0.63 \frac{J}{m^{3}}$

Step by step solution

01

Given

i) Radius of the circular loop $R = 50 mm = 50 脳 10^{- 3} m$

ii) Current in the circular loop $i = 100 A$

02

Understanding the concept

We can use the formula of magnetic field at the center of the current loop to find it. Using this magnetic field into the formula of the energy density at that point, we can find the energy density at the center of the loop.

Formula:

$B = 渭_{0} i / 2 R u_{B} = B^{2} / 2 渭_{0}$

03

(a) Calculate the magnetic field strength

The magnetic field at the center of loop is given by

$B = \frac{渭_{0} i}{2 R} B = \frac{(4 脳 10^{- 7} \frac{H}{m}) (100 A)}{2 (50 脳 10^{- 3} m)} B = 1.3 脳 10^{- 3} T$

Therefore, the magnetic field strength is $1.3 脳 10^{- 3} T$ .

04

(b) Calculate the energy density at the center of the loop

The energy density at the centre of the loop is given by

$u_{B} = \frac{B^{2}}{2 渭_{0}} u_{B} = \frac{{(1.3 脳 10^{- 3} T)}^{2}}{2 (4 蟺脳 10^{- 7} \frac{H}{m})} u_{B} = 0.63 \frac{J}{m^{3}}$

Therefore, energy density at the center of the loop is role="math" localid="1661424939827" $0.63 J / m^{3}$ .

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Most popular questions from this chapter

In Figure, $R = 15 惟, L = 5.0 H$ the ideal battery has $蔚 = 10 V$ , and the fuse in the upper branch is an ideal3.0 A fuse. It has zero resistance as long as the current through it remains less than3.0 A . If the current reaches3.0 A , the fuse 鈥渂lows鈥� and thereafter has infinite resistance. Switch S is closed at timet = 0 . (a) When does the fuse blow? (Hint: Equation 30-41 does not apply. Rethink Eq. 30-39.) (b) Sketch a graph of the current i through the inductor as a function of time. Mark the time at which the fuse blows.

At $t = 0$ , a battery is connected to a series arrangement of a resistor and an inductor. If the inductive time constant is 37.0 ms, at what time is the rate at which energy is dissipated in the resistor equal to the rate at which energy is stored in the inductor鈥檚 magnetic field?

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