A charge of -8.3 μC is traveling at a speed of 7.4 ×10^6 m / s in a region of space where there

A charge of -8.3 μC is traveling at a speed of 7.4 ×10^6 m /...

A charge of $-8.3 \mu \mathrm{C}$ is traveling at a speed of $7.4 \times 10^{6} \mathrm{m} / \mathrm{s}$ in a region of space where there is a magnetic field. The angle between the velocity of the charge and the field is $52^{\circ} .$ A force of magnitude $5.4 \times 10^{-3} \mathrm{N}$ acts on the charge. What is the magnitude of the magnetic field?

Key Concepts

Magnetic Force on a Moving Charge

This concept is based on the Lorentz force law, which states that a charged particle moving in a magnetic field experiences a force perpendicular to both its velocity and the field. The magnitude of the force is given by the equation F = |q| v B sin(?), where q is the charge, v is the speed of the particle, B is the magnetic field strength, and ? is the angle between the velocity vector and the magnetic field vector.

Vector Cross Product and Trigonometry

The cross product in vector mathematics is used to determine the direction and magnitude of the force acting on a charged particle in a magnetic field. The sine function in the equation represents the role of the angle between the particle's velocity and the magnetic field, emphasizing the contribution of both magnitude and direction in the interaction.

Unit Conversion in Electromagnetism

Proper conversion of units is critical in physics to ensure accuracy in calculations. In electromagnetic problems, charges may be given in units such as microcoulombs which need to be converted to the standard SI unit, coulombs, so that they are consistent with other quantities like velocity (meters per second) and magnetic field strength (tesla) used in the force equation.

Transcript

00:02 Okay, for this problem, the equation that we're going to want to use is magnetic field b is equal to force divided by charge times the velocity, sign of the angle between the velocity and the force.

00:39 Everything that we are given here is the charge.

00:45 The charge is minus 8 .3 microculums.

01:02 So point of note, microculum is the same thing as writing, you can take this micro here and just substitute it for times 10 to the minus 6th.

01:13 That's what a micro means.

01:20 Very small number.

01:23 And that's coolums instead of microculums.

01:26 So we'll want this in cool, so we'll want to include this times 10 to the minus 6th.

01:31 We are also given the velocity.

01:39 Velocity is equal to 7 .4 times 10 to the 6th.

01:53 It's a positive 6th.

01:55 And our units of velocity are in meters per second.

02:01 We are also given the angle between them, theta, is equal to 52 degrees.

02:16 The last piece of information we have is what the force is.

02:21 Our force is equal to 5 .4 times 10 to the minus 3.

02:59 And force, as usual, is in units of newton's.

03:04 Okay, let's look back at our equation.

03:08 We have force, we have charge, velocity, and we have our angle.

03:11 That's everything that we're going to need.

03:13 Okay, so we can plug all of that.

03:15 In straight into the formula take sign of the 52 multiply that by your velocity 7 .4 times q and then divide force by that number and what you'll end up with is b is equal to 1 .1 times 10 to the minus 4 all right we need our units now let's look at what we have so we have force and the top, newtons, divided by charge, coolum, multiplied by velocity meters per second.

04:09 And we know that a tesla is a newton second divided by a coolum meter.

04:28 Okay, so let's see if we can get this to look like that...

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