In Fig. 22-54, a nonconducting rod of length L= 8.15 cm has a charge -q=-4.23 fC uniformly dist

In Fig. 22-54, a nonconducting rod of length L= 8.15 cm has...

In Fig. $22-54$, a nonconducting rod of length $L=$ $8.15 \mathrm{~cm}$ has a charge $-q=-4.23 \mathrm{fC}$ uniformly distributed along its length. (a) What is the linear charge density of the rod? What are the (b) magnitude and (c) direction (relative to the positive direction of the $x$ axis) of the electric field produced at point $P$, at distance $a=12.0 \mathrm{~cm}$ from the rod? What is the electric field magnitude produced at distance $a=50 \mathrm{~m}$ by (d) the rod and (e) a particle of charge $-q=-4.23 \mathrm{fC}$ that we use to replace the rod? (At that distance, the rod "looks" like a particle.

In Fig. $22-54$, a nonconducting rod of length $L=$ $8.15 \mathrm{~cm}$ has a charge $-q=-4.23 \mathrm{fC}$ uniformly distributed along its length.
(a) What is the linear charge density of the rod? What are the
(b) magnitude and
(c) direction (relative to the positive direction of the $x$ axis) of the electric field produced at point $P$, at distance $a=12.0 \mathrm{~cm}$ from the rod? What is the electric field magnitude produced at distance $a=50 \mathrm{~m}$ by
(d) the rod and
(e) a particle of charge $-q=-4.23 \mathrm{fC}$ that we use to replace the rod? (At that distance, the rod "looks" like a particle.

Key Concepts

Direction of the Electric Field

The direction of the electric field produced by a charge depends on the sign of the charge: it points away from positive charges and toward negative charges. In vector analysis, the chosen coordinate system sets the positive direction. Understanding this principle is essential for determining the orientation of the field vectors relative to the reference axis in problems involving electrostatics.

Electric Field due to a Continuous Charge Distribution

The electric field generated by a continuous charge distribution, such as a uniformly charged rod, is calculated by integrating the contributions from all infinitesimal elements of charge along the object. Each element contributes to the field at a given point according to Coulomb's law, and the net field is the vector sum of these contributions. This approach is essential for non-point charge configurations.

Electric Field due to a Point Charge

For point charges, the electric field at a distance is described by Coulomb's law, which states that the magnitude of the electric field is directly proportional to the charge and inversely proportional to the square of the distance from the charge. This concept is foundational in electrostatics and is used to compare or approximate the fields produced by extended charge distributions at large distances.

Uniform Charge Distribution

A uniformly charged distribution means that the charge is spread evenly over the length (or area or volume) of an object. In the context of a rod or line charge, this implies that every segment of the rod contains the same amount of charge per unit length, which simplifies the computation of electric fields because the charge density remains constant along its length.

Linear Charge Density

Linear charge density is defined as the charge per unit length along a one-dimensional object, such as a rod. Mathematically, it is given by the total charge divided by the length over which it is distributed. This concept is crucial for analyzing systems where charges are spread out continuously rather than concentrated at a single point.

Transcript

00:01 Okay, so we have a non -conducting rod of length l, and it has a charge negative q uniformly distributed upon its length.

00:10 So we're going to put that on a coordinate system, and from 0 to l, this is our rod, and it is negatively charged.

00:23 Okay, so let's write down our givens to begin with.

00:28 We are given that l is 8 .15 centen.

00:33 Meters, we need to put this in si unit, so it's 0 .0 815 meters.

00:41 And you're given that negative q is negative 4 .23 times 10 to the negative 15 colognes.

01:01 So part a is we need to find the linear charge density of the rod.

01:09 Let's recall that the linear charge density is denoted by lambda, so lambda is equal to, in this case, negative q over l.

01:22 Plugging in the given values for q and l that we just discussed, you get negative 5 .19 times 10 to the negative 14 colognes per meter.

01:39 This is your answer for part a.

01:44 Okay, so part b, it wants the magnitude of the electric field produced at point p a distance a from the rod.

01:57 And it gives us a value for a, but let's just discuss this in general first.

02:03 So part b, let's go and look at our diagram one more time.

02:12 So it wants us to look at the electric field at point p, and it says it's a distance a from the rod.

02:22 So this point here is going to be l plus a.

02:26 And it wants the electric field at that point.

02:29 Remember that this is a rod of charge.

02:32 So for us to find the electric field, let's find it in this small infinitesimal area.

02:42 So this is dx.

02:44 And then we can integrate and find it over the whole rod.

02:50 So let's also notice that we are calling this our positive x -axis.

02:57 So the electric field is going to be along the x -axis, and there is no y component.

03:06 So we don't have to worry about the y component.

03:11 Okay, so let us find the magnitude of the electric field.

03:15 We're actually going to find the electric field how it is, and then we're going to determine the magnitude in the direction from that.

03:21 You'll understand what i'm saying in just a second.

03:23 So dx is the infinitesimal length of the rod.

03:27 It is at, of course, since we called it dx, this is x.

03:34 So that is x, i point dx.

03:39 And we know that dq is lambda dx, since lambda is equal to dq over dx as we said before so the electric field this small component dq this can be thought of as a point charge so we can use our equation for a point charge so this is the infinitesimal portion dx it's one over four pi epsilon not and then q, dq, which is lambda dx, over the distance squared.

04:44 So let's look at what this distance should be.

04:49 So this distance needs to be the distance from the point, from the infinitesimal point to p.

05:02 So you need this distance...

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