A charge of 28.0 nC is placed in a uniform electric field...

A charge of 28.0 $\mathrm{nC}$ is placed in a uniform electric field that is directed vertically upward and that has a magnitude of $4.00 \times 10^{4} \mathrm{N} / \mathrm{C}$ . What work is done by the electric force when the charge moves (a) 0.450 $\mathrm{m}$ to the right; (b) 0.670 $\mathrm{m}$ upward; (c) 2.60 $\mathrm{m}$ at an angle of $45.0^{\circ}$ downward from the horizontal?

A charge of 28.0 $\mathrm{nC}$ is placed in a uniform electric field that is directed vertically upward and that has a magnitude of $4.00 \times 10^{4} \mathrm{N} / \mathrm{C}$ . What work is done by the electric force when the charge moves (a) 0.450 $\mathrm{m}$ to the right; (b) 0.670 $\mathrm{m}$
upward; (c) 2.60 $\mathrm{m}$ at an angle of $45.0^{\circ}$ downward from the
horizontal?

Key Concepts

Work in Electric Fields

This concept involves calculating the work done on a charged particle within an electric field. The work is determined by the product of the charge, the magnitude of the electric field, and the displacement component in the direction of the field. This is mathematically expressed using the dot product of the electric field vector and the displacement vector.

Dot Product in Work Calculations

The dot product is a mathematical operation used to project one vector onto another. In the context of work done by an electric force, it identifies the portion of the displacement that aligns with the electric field direction, which is crucial because only this component contributes to the work done.

Uniform Electric Field

A uniform electric field is characterized by having the same magnitude and direction throughout the region. Such a field simplifies the analysis because the force acting on a charge remains constant regardless of the charge's position, allowing for straightforward work calculations using the formula work = qE·d.

Vector Projection

Vector projection involves decomposing a displacement vector into components along specific directions. For work calculations in electric fields, it is important to extract the displacement component parallel to the field vector, since this determines the actual work performed by the field on the moving charge.

Influence of Displacement Direction

The direction of displacement relative to the direction of the electric field is critical in determining the work done. If the displacement is perpendicular to the electric field, no work is done; if it is parallel, the full force contributes to the work; and for any oblique angle, only the parallel component affects the work.

Transcript

00:01 So for this problem, we have an electric field that's going upward, and it has a magnitude of 4 .00 times 10 to the 4 newtons per coulum.

00:15 And the charge of the charge in the field is 28 times 10 to the minus 9 couloms.

00:28 And our goal is to get the work for a few different paths.

00:31 So for a the path is this.

00:37 It goes to the right.

00:40 Let's see, what does it say? 0 .45 meters to the right.

00:45 And so we want to find the work by the electric force.

00:50 So work is equal to force dotted into displacement.

00:56 And so the force is directed up and then displacement is directed to the right.

01:03 And so because there's no component of the force that's parallel to the displacement, the work is zero.

01:12 And also to be clear, the force is the electric force, and that's going to be this qe.

01:20 So we're dotting the electric field, which is pointing up into the displacement, which is pointing to the right, and then that because they're perpendicular, that's zero.

01:29 Another way to think about it is you could take the cosine of the angle between e and d because you could, it's technically also q, e, d, cosine of the angle between them.

01:42 So here's the angle between them.

01:45 And it's 90 degrees.

01:47 And so that becomes zero.

01:49 And so for b, the path is up the along the electric field.

01:57 And so this time there is a component of the force along the electric, or excuse me, along the displacement...

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