(a) What is the force per meter on a lightning bolt at the equator that carries 20,000 A perpendicular to Earth’s $3.0 \times 10 ^ { - 5 } \mathrm { T }$ field? (b) What is the direction of the force if the current is straight up and Earth’s field direction is due north, parallel to the ground?

(a) A cosmic ray proton moving toward Earth at $5.00 \times 10 ^ { 7 } \mathrm { m } / \mathrm { s }$ experiences a magnetic force of $1.70 \times 10 ^ { - 16 } \mathrm { N }$. What is the strength of the magnetic field if there is a $45 ^ { \circ }$ angle between it and the proton’s velocity? (b) Is the value obtained in part a. consistent with the known strength of Earth’s magnetic field on its surface? Discuss.

$(a)$ What is the force per meter on a lightning bolt at the equator that carries $20,000 \mathrm{~A}$ perpendicular to Earth's $3.0 \times 10^{-5} \mathrm{~T}$ field? $(b)$ What is the direction of the force if the current is straight up and Earth's field direction is due north, parallel to the ground?

(a) By how many percent is the torque of a motor decreased if its permanent magnets lose 5.0% of their strength? (b) How many percent would the current need to be increased to return the torque to original values?

(a) What voltage will accelerate electrons to a speed of $6.00 \times 10 ^ { - 7 } \mathrm { m } / \mathrm { s }$? (b) Find the radius of curvature of the path of a proton accelerated through this potential in a 0.500-T field and compare this with the radius of curvature of an electron accelerated through the same potential.

A cosmic-ray electron moves at $7.5 \times 10 ^ { 6 } \mathrm { m } / \mathrm { s }$ perpendicular to Earth’s magnetic field at an altitude where the field strength is $1.0 \times 10 ^ { - 5 } \mathrm { T }$. What is the radius of the circular path the electron follows?

An alpha-particle $\left( m = 6.64 \times 10 ^ { - 27 } \mathrm { kg }, q = 3.2 \times 10 ^ { - 19 } \mathrm { C }\right )$ travels in a circular path of radius 25 cm in a uniform magnetic field of magnitude 1.5 T. (a) What is the speed of the particle? (b) What is the kinetic energy in electron-volts? (c) Through what potential difference must the particle be accelerated in order to give it this kinetic energy?

(a) What voltage will accelerate electrons to a speed of $6.00 \times 10^{-7}\ \mathrm{m} / \mathrm{s} ?$ (b) Find the radius of curvature of the path of a proton accelerated through this potential in a 0.500-T field and compare this with the radius of curvature of an electron accelerated through the same potential.

A strip of copper is placed in a uniform magnetic field of magnitude 2.5 T. The Hall electric field is measured to be $1.5 \times 10 ^ { - 3 } \mathrm { V } / \mathrm { m }$. (a) What is the drift speed of the conduction electrons? (b) Assuming that $\mathrm { n } = 8.0 \times 10 ^ { 28 }$ electrons per cubic meter and that the cross-sectional area of the strip is $5.0 \times 10 ^ { - 6 } \mathrm { m } ^ { 2 }$, calculate the current in the strip. (c) What is the Hall coefficient 1/nq?

(a) A dc power line for a light-rail system carries 1000 A at an angle of $30.0 ^ { \circ }$ to Earth’s $5.0 \times 10 ^ { - 5 } \mathrm { T }$ field. What is the force on a 100-m section of this line? (b) Discuss practical concerns this presents, if any.

In Exercise, the speed $\| \mathbf { v } \|$ of a particle at an arbitrary time $t$ is given. Find the scalar tangential component of acceleration at the indicated time.

$$\| \mathbf { v } \| = \sqrt { t ^ { 2 } + e ^ { - 3 t } } ; t = 0$$

Solve the inequality. Check your solution. $\frac{x}{x-3}>\frac{4}{x-2}$

In Exercise, a function $y = f ( x )$ and values of $x _ { 0 }$ and $x_ { 1 }$are given. (a) Fird the average rate of change of y with respect to x over the interval $\left[ x _ { 0 } , x _ { 1 } \right]$ (b) Find the instantaneous rate of change of y with respect to x at the given value of $x _ { 0 . }$ (c) Find the instantaneous rate of change of y with respect to x at a general point $x _ { 0 }$ . (d) Sketch the graph of $y = f ( x )$ logether with the secant and tangent lines whose slopes are given by the results in parts (a) and (b). $y = x ^ { 3 } ; x _ { 0 } = 1 , x _ { 1 } = 2$

Inside a motor, 30.0 A passes through a 250-turn circular loop that is 10.0 cm in radius. What is the magnetic field strength created at its center?