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Chapter 29: Q.67 (page 834)

If A particle of charge $q$ and mass $m$ moves in the uniform fields $\overset{鈫�}{E} = E_{0} \hat{k} and \overset{鈫�}{B} = B_{0} \hat{k}$ . At $t = 0$ , the particle has velocity ${\overset{鈫�}{v}}_{0} = v_{0} \hat{i}$ . What is the particle's speed at a later time $r$ ?

Short Answer

Expert verified

The particle's speed at a later time $v = \sqrt{v_{o}^{2} + {(\frac{q E_{o} t}{m})}^{2}}$

Step by step solution

01

Given Values

$\overset{鈫�}{E} = E_{o} \hat{k}$

$\overset{鈫�}{B} = B_{o} \hat{k}$

$\overset{鈫�}{v} = v_{o} \hat{i}$

$t = 0$

We use the equations,

${\overset{鈫�}{F}}_{E} = q \overset{鈫�}{E} = q E_{o} \hat{k}$

$\overset{鈫�}{F_{M}} = \overset{鈫�}{q v} 脳 \overset{鈫�}{B} = q v_{o} B_{o} (鈭� j)$

In the x-y plane, for circular movement:

02

Equation Solving

$\overset{鈫�}{F_{M}} = q v_{o} B_{o} \hat{r} 鉄� F_{E} = q E_{o} \hat{k}$

$鈭� \overset{鈫�}{F} = m \overset{鈫�}{a}$

$F_{1 z} = q E_{o} 鈭� m a_{z}$

$F_{1 r} = q v_{o} B_{o} = m a_{蟿} = \frac{m v_{o}^{2}}{2}$

$F_{1 t} = 0$

03

Final Equation

$we find \overset{鈫�}{v} (t) :$

$\overset{鈫�}{v} (t) = v_{o} \hat{r} + (0 + a_{z} t) \hat{k}$

$\overset{鈫�}{v} (t) = v_{o} \hat{i} + a_{z} t \hat{k}$

$v = \sqrt{v_{o}^{2} + {(a_{z} t)}^{2}}$ $(a_{z} = \frac{q E_{o}}{m} 鈰�)$

$v = \sqrt{v_{o}^{2} + {(\frac{q E_{o} t}{m})}^{2}}$

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Most popular questions from this chapter

The on-axis magnetic field strength $10 cm$ from a small bar magnet is $5.0 渭罢$ .

a. What is the bar magnet鈥檚 magnetic dipole moment?

b. What is the on-axis field strength 15 cm from the magnet?

The right edge of the circuit in FIGURE EX29.35 extends into a $50 m T$ uniform magnetic field. What are the magnitude and direction of the net force on the circuit?

The magnetic field strength at the north pole of a $2.0 c m$ - diameter, $8 c m$ -long Alnico magnet is $0.10 T$ . To produce the same field with a solenoid of the same size, carrying a current of $2.0 A$ , how many turns of wire would you need?

The earth鈥檚 magnetic dipole moment is $8.0 脳 10^{22} A . m^{2}$ .

a. What is the magnetic field strength on the surface of the earth at the earth鈥檚 north magnetic pole? How does this compare to the value in Table 29.1? You can assume that the current loop is deep inside the earth.

b. Astronauts discover an earth-size planet without a magnetic field. To create a magnetic field at the north pole with the same strength as earth鈥檚, they propose running a current through a wire around the equator. What size current would be needed?

What are the strength and direction of the magnetic field at the center of the loop in FIGURE P29.43?

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