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physics
university physics

Questions and Answers of University Physics

In Example 31.6 (Section 31.4), a hair dryer is treated as a pure resistor. But because there are coils in the heating element and in the motor that drives the blower fan, a hair dryer also has
The discussion of magnetic forces on current loops in Section 27.7 commented that no net force is exerted on a complete loop in a uniform magnetic field, only a torque. Yet magnetized materials that
If the battery in Discussion Question Q26.10 is ideal with no internal resistance, what will happen to the brightness of the bulb when S is closed? Why?
In the conical pendulum in Example 5.20 (Section 5.4), which of the forces do work on the bob while it is swinging?In Example 5.20,An inventor designs a pendulum clock using a bob with mass m at the
Let $G$ be the group of discrete transformations that leave a rectangle invariant (with the composition law given by subsequent application of two transformations as the group product), including the
Let $S^{2}$ denote a sphere of unit radius, centered at the origin of $\mathbb{R}^{3}$. What is the group of transformations that leave it invariant? Next, let $S_{(1)}^{2}$ be a sphere of unit
Let $S_{(2)}^{2}$ denote a sphere of unit radius, centered at the origin of threedimensional space, with two noncoincident points on its surface. What is the group of transformations that leave
Is $\mathrm{O}(2)$ an Abelian or a non-Abelian group?
Prove that the order of the symmetric group of a set of $N$ elements, $S_{N}$, is $N$ !.
Prove that the only finite groups of order 4 are $\mathbb{Z}_{2} \times \mathbb{Z}_{2}$ and $\mathbb{Z}_{4}$.
Prove that the cyclic group of order 3 does not have proper subgroups.
Prove that the groups $(\operatorname{Mat}(n, \mathbb{R}),+)$ and $\left(\mathbb{R}^{n^{2}},+\right)$ are isomorphic.
Let $N$ be a positive integer. Consider the relation $\circledast$ among pairs of integers $r, s \in \mathbb{Z}$ defined as $r \circledast s$ when $r-s$ is an integer multiple of $N$. Prove that
Consider the sets and the composition laws that are defined as follows:- The set of $2 \times 3$ matrices with real coefficients, and the composition law defined as the entry-by-entry matrix sum.-
Define three different functions $f_{i}: \mathbb{R} \rightarrow \mathbb{R}, i \in\{1,2,3\}$ in such a way that- $f_{1}$ is an injection but not a surjection,- $f_{2}$ is a surjection but not an
Given three generic functions $f_{i}: \mathbb{R} \rightarrow \mathbb{R}, i \in\{1,2,3\}$, such that $f_{1}$ is an injection but not a surjection, $f_{2}$ is a surjection but not an injection, and
Consider the group of permutations of the set $X=\{1,2,3,4,5,6\}$. Compute the following products, where permutations are denoted using the shortened cycle notation, i.e., omitting one-element
Let $X$ be the $n$-dimensional unit sphere $S^{n}=\left\{x \in \mathbb{R}^{n+1}:|x|=1\right\}$ with antipodal points identified, and let $A$ be the $(n+1)$-dimensional real vector space without the
Consider the two groups $G_{1}=\mathrm{SO}(3)$ and $G_{2}=\mathrm{U}(1)$ and a surjective mapping $\mu: G_{1} \rightarrow G_{2}$. Discuss whether, in general, $\mu$ is a homomorphism, or not.
Let $x$ and $y$ be two elements of the five-dimensional complex vector space $\mathbb{C}^{5}$, and let $A$ be a set, such that each of its elements can be written as $(\alpha x+\beta y)$, with
The default privacy settings of a certain online social networking service are such, that messages posted by a user are visible to their author and to all of her/his contacts, as well as to the
Given a group $G$, a left $G$-space $X$, and an element $x \in X$, prove that the isotropy group of $x$ is a subgroup of $G$.
Given the set $X=\{1,2,3,4,5\}$, construct the inverse of each of the following elements of $\operatorname{Perm}(X)$ :- $(24)(35)$,- $(14235)$,- $(1)(2)(34)$,- $(253)$,- $(15)(24) \cdot(154)(23)$.
Given two vector spaces $V_{1}$ and $V_{2}$, prove that the dimension of their direct sum is $\operatorname{dim}\left(V_{1} \oplus V_{2}\right)=\operatorname{dim} V_{1}+\operatorname{dim} V_{2}$.
Given a unitary representation of a group on the vector space $V$, in which a scalar product $\langle\ldots \mid \ldotsangle$ is defined, and given a submodule $W$, and its orthogonal complement
Given a finite group $G$, prove that the matrices of its left-regular representation, with elements defined by Eq. (3.70), satisfy the group multiplication law, i.e., that $L(g)
Given a finite group $G$, prove that the matrices of its right-regular representation, defined according to Eq. (3.73), obey the group multiplication law, i.e., $R(g)
Given a finite group $G$, prove that its left-regular and right-regular representations are isomorphic to each other.
Given a vector space $V$, prove that every $\omega \in\left(V^{\star}\right)^{\star}$ can be uniquely associated with a vector $\vec{v} \in V$, such that $\omega(f)=\langlef, \vec{v}angle$.
Consider a linear operator $A$ acting on a vector space $V$ of finite dimension $N$ and a linear operator $B$ acting on a vector space $W$ of finite dimension $M$. Assuming that orthonormal bases
Show that $\mathbb{R}^{n}$ with the usual topology is a Hausdorff space.
Consider $\mathbb{R}^{2}$ equipped with the discrete metric$$d(x, y)= \begin{cases}1 & \text { if } x eq y \\ 0 & \text { if } x=y\end{cases}$$where $x$ and $y$ are elements of $\mathbb{R}^{2}$, and
Let $f: M \rightarrow N$ be a homeomorphism. Define a map $f_{\star}: \pi_{1}\left(M, x_{0}\right) \rightarrow$ $\pi_{1}\left(N, f\left(x_{0}\right)\right)$ such that $f_{\star}([\gamma])=[f \circ
Show that the homeomorphism between topological spaces is an equivalence relation.
Prove that, if $M$ is an $n$-dimensional manifold with boundary, $\partial M$ is an ( $n-1)$ dimensional manifold.
Given the subset of $\mathbb{R}^{2}$ defined as follows:$$S=\left\{(x, y) \in \mathbb{R}^{2} \mid y=\sin (1 / x), x>0\right\} \cup\{(0,0)\}$$prove that its subset $S_{+}=\left\{(x, y) \in
This problem focuses on the determination of homotopy groups.(i) Let $M=\mathbb{R}^{3} \backslash\{$ a point $\}$. Find $\pi_{1}(M)$ and $\pi_{2}(M)$.(ii) Let $M=\mathbb{R}^{3} \backslash\{$ a line
Find the fundamental homotopy group of a torus with one point removed, $T^{2} \backslash\{$ a point $\}$. Hint: It may be useful to think of a torus as a rectangle with opposite sides identified.
Find an atlas and coordinates for a torus $T^{2}=S^{1} \times S^{1}$.
Let a differentiable manifold $M_{1}$ be the set of real numbers $\mathbb{R}$, with the coordinate $\phi_{1}(x)=x$, and $M_{2}$ also $\mathbb{R}$ but with the coordinate $\phi_{2}(x)=x^{3}$. Show
Derive the transformation rule for the components of the tensor$$T=T^{\mu_{1} \mu_{2}}{ }_{v_{1} v_{2} v_{3}} \frac{\partial}{\partial x^{\mu_{1}}} \otimes \frac{\partial}{\partial x^{\mu_{2}}}
Let the $(1,0)$-tensor$$R=\sum_{\mu=1}^{3} R^{\mu} \frac{\partial}{\partial x^{\mu}}$$have the components$$R^{1}=a, \quad R^{2}=a^{2}, \quad R^{3}=a^{3},$$and let the
In classical mechanics the equations of motion for a system with $n$ degrees of freedom can be written as a set of first-order differential equations$$\frac{d q_{i}}{d t}=\frac{\partial H}{\partial
Show that the Leibniz rule expressed by Eq. (4.111) is satisfied when $T_{1} \equiv f$ is a function, and $T_{2}$ is a vector field $X$ or a one-form $\omega$.Data from Eq. (4.111) Lx(T1 T2) (LxT)
Let $T$ be a $(p, q)$-tensor field, and let $X$ and $Y$ be vector fields. Show that the Lie derivative satisfies $\left[\mathcal{L}_{X}, \mathcal{L}_{Y}\right] T=\mathcal{L}_{[X, Y]} T$. It is enough
Let $X=X^{\mu}(x) \frac{\partial}{\partial x^{\mu}}$ be a vector field and let $g=g_{\mu v}(x) d x^{\mu} \otimes d x^{v}$ be a $(0,2)$ tensor. Compute the Lie derivative $\mathcal{L}_{X} g$.
Given the $q$-form $\omega$, the $r$-form $\eta$, and the $s$-form $\xi$, show that the exterior product satisfies the properties:(i) $\omega \wedge \omega=0$, when the order of $\omega$ is odd,(ii)
Show that under a coordinate transformation $\left(x_{1}, \ldots, x_{n}\right) \rightarrow\left(y_{1}(\vec{x}), \ldots, y_{n}(\vec{x})\right)$ in $\mathbb{R}^{n}$, the $n$-form transforms as$$d y^{1}
Show that the exterior derivative of the pullback of a form equals the pullback of the exterior derivative: $d f^{\star} \omega=f^{\star}(d \omega)$.
Let $M$ be the four-dimensional Minkowski space, with coordinates $x^{0}, x^{1}, x^{2}$, and $x^{3}$. Let us define a linear operator $*: \Omega^{r}(M) \rightarrow$ $\Omega^{4-r}(M)$, such
Consider the following differential forms in $\mathbb{R}^{3}$ :$$\alpha=x d x+y d y+z d z, \quad \beta=z d x+x d y+y d z, \quad \gamma=x y d z$$(i) Is $\alpha$ closed or exact? Is $\gamma$ closed or
Examine if the following differential forms are closed, explaining why:(i) $A \in \Omega^{1}\left(\mathbb{R}^{2} \backslash\{0\}\right)$, defined as$$A=\frac{h c}{2 \pi e} \frac{x d y-y d
Let $X$ and $Y$ be vector fields, let $\omega$ be a $r$-form, and let $\eta$ be an $s$-form. Show that the following properties of the interior product are true:(i) $i_{X} i_{Y} \omega=-i_{Y} i_{X}
Consider the vector field $X=x \partial_{x}+y \partial_{y}+z \partial_{z}$, the two-form $\alpha=2 z d x \wedge d y+$ $3 y d x \wedge d z$, and the three-form $\omega=d x \wedge d y \wedge d z$ on
Given the two-dimensional torus $T^{2}$, parameterized in terms of the two real variables $x^{1} \in[0,2 \pi)$ and $x^{2} \in[0,2 \pi)$, consider its embedding into the threedimensional Euclidean
Let $\theta$ and $\phi$ be the polar coordinates. Introduce the complex numbers $z$ and $\bar{z}$, where$$\begin{equation*}z=e^{i \phi} \tan (\theta / 2) \equiv \xi+i \eta
Let $X$ and $Y$ be vector fields and $f$ a function on $M$. Compute the "double covariant derivatives"(i) $abla_{X} abla_{Y}$,(ii) $abla_{\mu} abla_{v} f$, writing them as sums of terms that involve
Let $\Gamma_{\mu u}^{\alpha}$ be the Levi-Civita connection (implying the symmetry $\Gamma_{\mu u}^{\alpha}=\Gamma_{v \mu}^{\alpha}$ ).(i) Show that $abla_{\mu} abla_{v} f=abla_{v} abla_{\mu} f$.(ii)
Let the metric on a two-dimensional torus $T^{2}$ with radii $r$ and $R>r$ be$$\begin{equation*}g=r^{2} d \theta \otimes d \theta+(R+r \cos \theta)^{2} d \phi \otimes d \phi
Spatially homogeneous and isotropic universe. The Robertson-Walker metric$$\begin{equation*}g=-d t \otimes d t+a^{2}(t)\left(\frac{d r \otimes d r}{1-k r^{2}}+r^{2}\left(d \theta \otimes d
Show that, given two tangent vectors $V$ and $W$ of a hypersurface $\Sigma$, contraction with the projection tensor $P_{\mu u}$ reduces to a scalar product:$$\begin{equation*}P_{\mu u} V^{\mu}
Show that the scalar $K$, which, according to Eq. (5.366), is constructed from the extrinsic curvature as $K=g^{\mu v} K_{\mu u}$, is equal to the covariant divergence of the normal vector field,
Find all elements of the matrix $\exp (i \alpha A)$, where$$A=\left(\begin{array}{lll}0 & 0 & 1 \tag{6.407}\\0 & 0 & 0 \\1 & 0 & 0\end{array}\right)$$
Given two matrices $A$ and $B$ satisfying the commutation relation$$\begin{equation*}[A, B]=B \tag{6.408}\end{equation*}$$calculate$$\begin{equation*}\exp (i \alpha A) B \exp (-i \alpha A)
Let $D_{1}$ and $D_{2}$ be two irreducible representations of a Lie algebra, of dimensions $\operatorname{dim} D_{1}$ and $\operatorname{dim} D_{2}$, and with Dynkin indices $\lambda_{D_{1}}$ and
Given the generators $X_{a}$ (with $a=1, \ldots, N$ ) of a Lie algebra, prove that the second-order Casimir operator $C^{(2)}=\sum_{a=1}^{N} X_{a} X_{a}$ commutes with all the generators.
Starting from the relation $\left[H_{i}, E_{\alpha}\right]=\alpha_{i} E_{\alpha}$ and its adjoint, and using the Hermiticity of the Cartan generators, show that the $E_{\alpha} \mathrm{s}$ are
Show that $\left[E_{\alpha}, E_{\beta}\right]$ is proportional to $E_{\alpha+\beta}$. What if $\alpha+\beta$ is not a root?
Compute the structure constants $f_{147}$ and $f_{458}$ of the algebra of generators of the $\mathrm{SU}(3)$ group.
Show that $\lambda_{2}, \lambda_{5}$ and $\lambda_{7}$ generate an su(2) subalgebra of the su(3) Lie algebra.
su(2) representations. Decompose the following tensor products of spin representations into a direct sum of spin representations.(i) $1 \otimes \frac{1}{2}$(ii) $1 \otimes \frac{1}{2} \otimes 1$(iii)
Check that the fundamental roots $\mu^{i}$ satisfy$$2 \frac{\alpha^{i} \cdot \mu^{j}}{\left(\alpha^{i}\right)^{2}}=\delta^{i j}$$
Construct the Young tableaux of the following su(4) representations and compute their dimensions.(i) $(1,0,0)$(ii) $(1,1,1)$(iii) $(2,1,2)$
Decompose the following tensor product of su(5) irreducible representations. Check that the dimensions multiply/add up and match correctly.$$\begin{equation*}(0,0,1,0) \otimes(1,0,0,0)=?
The energy required to remove the 3s electron from a sodium atom in its ground state is about 5 eV. Would you expect the energy required to remove an additional electron to be about the same, or
(a) Can you show that the orbital angular momentum of an electron in any given direction (e.g., along the z-axis) is always less than or equal to its total orbital angular momentum? In which cases
Ionic bonds result from the electrical attraction of oppositely charged particles. Are other types of molecular bonds also electrical in nature, or is some other interaction involved? Explain.
In ionic bonds, an electron is transferred from one atom to another and thus no longer “belongs” to the atom from which it came. Are there similar transfers of ownership of electrons with other
Discuss the differences between the rotational and vibrational energy levels of the deuterium (“heavy hydrogen”) molecule D2 and those of the ordinary hydrogen molecule H2. A deuterium atom
For electronic devices such as amplifiers, what are some advantages of transistors compared to vacuum tubes? What are some disadvantages? Are there any situations in which vacuum tubes cannot be
Can you tell from the value of the mass number A whether to use a plus value, a minus value, or zero for the fifth term of Eq. (43.11)? Explain. (A – 2Z)² Z(Z – 1) C4 * C5A¬4/3 Ев — С1А
What are the main advantages of colliding-beam accelerators compared with those using stationary targets? What are the main disadvantages?
(a) If two electrons in hydrogen atoms have the same principal quantum number, can they have different orbital angular momentum? How?(b) If two electrons in hydrogen atoms have the same orbital
What are the most significant differences between the Bohr model of the hydrogen atom and the Schrödinger analysis? What are the similarities?
Find the width of a one-dimensional box for which the ground-state energy of an electron in the box equals the absolute value of the ground state of a hydrogen atom.
Find the wavelengths of a photon and an electron that have the same energy of 25 eV.
The peak-intensity wavelength of red dwarf stars, which have surface temperatures around 3000 K, is about 1000 nm, which is beyond the visible spectrum. So why are we able to see these stars, and why
The materials called phosphors that coat the inside of a fluorescent lamp convert ultraviolet radiation (from the mercury vapor discharge inside the tube) into visible light. Could one also make a
An ultrashort pulse has a duration of 9.00 fs and produces light at a wavelength of 556 nm. What are the momentum and momentum uncertainty of a single photon in the pulse?
A laser produces light of wavelength 625 nm in an ultrashort pulse. What is the minimum duration of the pulse if the minimum uncertainty in the energy of the photons is 1.0%?
The predominant sound waves used in human speech have wavelengths in the range from 1.0 to 3.0 meters. Using the ideas of diffraction, explain how it is possible to hear a person’s voice even when
In single-slit diffraction, what is sin(β/2) when In θ = 0? In view of your answer, why is the single-slit intensity not equal to zero at the center?
Devise straightforward experiments to measure the speed of light in a given glass using (a) Snell’s law; (b) Total internal reflection; (c) Brewster’s law.
According to Ampere’s law, is it possible to have both a conduction current and a displacement current at the same time? Is it possible for the effects of the two kinds of current to cancel each
A resistor, inductor, and capacitor are connected in parallel to an ac source with voltage amplitude V and angular frequency ?. Let the source voltage be given by ? = V cos ?t. (a) Show that the
A toroidal solenoid has 2900 closely wound turns, cross-sectional area 0.450 cm2, mean radius 9.00 cm, and resistance R = 2.80 Ω. The variation of the magnetic field across the cross section of the
A parallel-plate capacitor having square plates 4.50 cm on each side and 8.00 mm apart is placed in series with an ac source of angular frequency 650 rad s and voltage amplitude 22.5 V, a 75.0 Ω
(a) Show that for an L-R-C series circuit the power factor is equal to R/Z.(b) An L-R-C series circuit has phase angle –31.5°. The voltage amplitude of the source is 90.0 V. What is the voltage
Some electrical appliances operate equally well on ac or dc, and others work only on ac or only on dc. Give examples of each, and explain the differences
For Eq. (29.6), show that if ? is in meters per second, B in teslas, and L in meters, then the units of the right-hand side of the equation are joules per coulomb or volts (the correct SI units for

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