# 物理代写|PHYSICS 1 PHYS1001 University of Glasgow Assignment

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## Instructions:

Physics is a branch of science that deals with the study of matter, energy, and their interactions. It seeks to understand the fundamental laws of nature and the behavior of the universe at the most basic level. Physicists study everything from the tiniest subatomic particles to the largest structures in the universe, such as galaxies and black holes. They use mathematical models and experimental methods to develop theories and test them against observations. Physics plays a critical role in many areas of modern technology, including electronics, communications, and medicine. It has also contributed significantly to our understanding of the natural world and the universe we live in.

The line integral of a scalar function $f(x, y, z)$ along a path $C$ is defined as
$$\int_C f(x, y, z) d s=\lim {\substack{N \rightarrow \infty \ \Delta s_i \rightarrow 0}} \sum{i=1}^N f\left(x_i, y_i, z_i\right) \Delta s_i$$
where $C$ has been subdivided into $N$ segments, each with a length $\Delta s_i$. To evaluate the line integral, it is convenient to parameterize $C$ in terms of the arc length parameter $s$. With $x=x(s)$, $y=y(s)$ and $z=z(s)$, the above line integral can be rewritten as an ordinary definite integral:
$$\int_C f(x, y, z) d s=\int_{s_1}^{s_2} f[x(s), y(s), z(s)] d s$$

Yes, that is correct.

When we parameterize the curve $C$ using the arc length parameter $s$, we can express the coordinates of any point on the curve as functions of $s$ as $x(s)$, $y(s)$, and $z(s)$.

Using this parameterization, we can express the line integral as an ordinary definite integral over $s$ as shown above. The integrand $f[x(s), y(s), z(s)]$ represents the value of the scalar function $f$ evaluated at the point on the curve corresponding to the arc length parameter $s$.

The limit as $N\to\infty$ and $\Delta s_i\to 0$ in the definition of the line integral is necessary because we want to calculate the exact value of the integral, which involves summing up infinitely many infinitesimal contributions along the path $C$.

A solid cylinder of length $L$ and radius $R$, with $L \gg R$, is uniformly filled with a total charge $Q$. a. What is the volume charge density $\rho$ ? Check units!

a. The volume charge density $\rho$ is defined as the charge $Q$ per unit volume. In this case, the total charge $Q$ is uniformly distributed throughout the volume of the cylinder. The volume of a cylinder is given by $V=\pi R^2L$. Therefore, the volume charge density is:

$$\rho = \frac{Q}{V} = \frac{Q}{\pi R^2 L}$$

Checking units: $[Q]=C$, $[R]=m$, and $[L]=m$, so $[\rho]=\frac{C}{m^3}$.

b. Suppose you go very far away from the cylinder to a distance much greater than $R$. The cylinder now looks like a line of charge. What is the linear charge density $\lambda$ of that apparent line of charge? Check units!

b. When we are far away from the cylinder, it appears as a line of charge with linear charge density $\lambda$. The linear charge density is defined as the charge per unit length. We can find $\lambda$ by considering an element of length $dl$ on the cylinder. The charge on this element is $dq=\rho Adl$, where $A$ is the cross-sectional area of the cylinder (which is just a circle of radius $R$). The element of length $dl$ subtends an angle of $d\theta=\frac{dl}{R}$ at the observation point, and the distance from the element to the observation point is $r\gg R$. Therefore, the contribution to the electric field from this element is:

$$dE=\frac{1}{4\pi\epsilon_0}\frac{dq}{r^2}=\frac{1}{4\pi\epsilon_0}\frac{\rho Adl}{r^2}$$

The total electric field at the observation point is obtained by integrating over the entire length of the cylinder:

$$E=\int_{-L/2}^{L/2} dE = \int_{-L/2}^{L/2} \frac{1}{4\pi\epsilon_0}\frac{\rho A}{r^2}dl=\frac{\rho A}{4\pi\epsilon_0 r^2}\int_{-L/2}^{L/2}dl$$

The integral gives the total length of the cylinder, which is just $L$. Therefore, we have:

$$E=\frac{\rho AL}{4\pi\epsilon_0 r^2}$$

On the other hand, we know that the electric field due to a line of charge with linear charge density $\lambda$ is given by:

$$E=\frac{\lambda}{2\pi\epsilon_0 r}$$

Comparing the two expressions for $E$, we get:

$$\frac{\lambda}{2\pi\epsilon_0 r}=\frac{\rho AL}{4\pi\epsilon_0 r^2}$$

Solving for $\lambda$, we obtain:

$$\lambda=\frac{\rho AL}{2r}=\frac{Q}{2\pi R L}\qquad\text{(since }A=\pi R^2\text{)}$$

Checking units: $[Q]=C$, $[R]=m$, and $[L]=m$, so $[\lambda]=\frac{C}{m}$.