A positive charge q=3.20 ×10^-19 Cmoves with a velocity 𝐯=(2 𝐢̂+3 𝐣̂-𝐤̂) m / s through a region

A positive charge q=3.20 ×10^-19 Cmoves with a velocity 𝐯=(2...

A positive charge $q=3.20 \times 10^{-19} \mathrm{Cmoves}$ with a velocity $\mathbf{v}=(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-\hat{\mathbf{k}}) \mathrm{m} / \mathrm{s}$ through a region where both a uniform magnetic field and a uniform electric field exist. (a) Calculate the total force on the moving charge (in unit-vector notation), taking $\mathbf{B}=(2 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}+\mathbf{k}) \mathrm{T}$ and $\mathbf{E}=(4 \hat{\mathbf{i}}-\hat{\mathbf{j}}-2 \hat{\mathbf{k}}) \mathrm{V} / \mathrm{m} .$ (b) What angle does the force vector make with the positive $x$ axis?

Key Concepts

Vector Addition and Decomposition

This concept involves breaking down vectors into their components using unit vectors and then combining them to find a resultant vector. It is essential for analyzing forces in multiple dimensions, allowing for the precise calculation of net force on an object.

Angle Calculation in Vector Analysis

Angle calculation involves using trigonometric relationships to determine the orientation of a vector relative to a coordinate axis. By examining the components of a force vector, one can compute the angle that it makes with a specified axis, which is important for understanding the directionality of the resultant force.

Vector Cross Product

The vector cross product is a mathematical operation on two vectors in three-dimensional space that results in a third vector, perpendicular to the plane containing the original vectors. It is crucial in computing the magnetic force where direction and the sense of the force are determined.

Electric Force

Electric force is the force exerted on a charge due solely to an electric field, described by F_E = qE. It acts in the direction of the electric field for a positive charge and its magnitude depends directly on the magnitude of the charge and the electric field strength.

Lorentz Force

The Lorentz force represents the total force on a charged particle moving in electromagnetic fields, and is given by the equation F = q(E + v x B). This concept is fundamental in understanding how charged particles interact with both electric and magnetic fields simultaneously.

Magnetic Force

Magnetic force is the force acting on a moving charge in a magnetic field, calculated by F_B = q(v x B). It is always perpendicular to both the velocity of the charge and the magnetic field, and it results from the cross product of the two vectors.

Transcript

00:01 Question number 57 wants us to find the total force on a charge moving through a uniform magnetic field where a uniform electric field is also present.

00:11 The charge q, which is moving through these two combined fields, has a value of 3 .2 times 10 to the minus 19 coulums, and it's moving at a velocity v, given to us in vector notation, as 2 ihat plus 3 .3 .3.

00:30 3j hat minus k hat meters per second.

00:34 The magnitude and direction of the magnetic field the charge passes through is also given to us in vector notation.

00:41 It's equivalent to 2i -hat plus 4j -hat plus k -hat tesla's.

00:49 The magnitude and direction of the electric field also present is 4 i -hat minus j -hat, minus 2 k -hat volts per meter.

01:02 So everything in this problem, with the exception of the charge, has magnitude and direction to it.

01:08 The problem wants us to find the total force on the charge.

01:13 To find the total force, we'll need to have the combined sum of the force from the magnetic field, plus the force from the electric field put together.

01:25 Let's first work out what the magnetic force is.

01:29 F sub b, will be equal to qv cross b.

01:34 Let's write out the matrix for that cross multiplication product shall we? we'll have ijk in the top row followed by two three minus one representing the vector v in the second row and two for one representing the vector b in the bottom row.

01:57 Now in writing out the cross product, we can just have q outside of the brackets which will contain the rest of the solution, which will be multiplied by q.

02:10 And for our cross product, we'll have three times one minus four times negative one times i hat, minus two times one, minus two times negative one, j hat, plus two times 4 minus 2 times 3 k hat.

02:31 Then working out all the multiplication in the parentheses in front of the vector components we'll have q times 3 minus minus 4 i hat minus 2 minus 2 j hat plus 8 minus 6 k hat.

02:50 Finally when we take the sums within all the parentheses in front of the vector components.

02:57 Our expression for f sub b will be cube times 7 i hat minus 4 j hat plus 2 k hat and the units on that will be in newton's per coolum.

03:11 So now we have our f sub b vector but what about our f sub e vector.

03:18 In our chapter on electric fields we learned that the electric field vector was equal to f sub e, the force of the electric field, divided by the charge q...

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