Recall that angular momentum, , is equal to an object's moment of inertia multiplied by its angular velocity, . This means that if we can solve for and for , we'll be able to compute . As far as the moment of inertia, when we look up in a table the value for a uniform sphere, that's equal to two-fifths the mass of the sphere multiplied by the square of its radius the formula for angular momentum is L=Iw. L-angular momentum. I-moment of inertia. w-angular velocity (r*v in your case) moment of inertia for a uniform sphere is (2/5)m*r^2. I think you can do the rest. Dec 27, 2003 welcome back to camp 131 a today were gonna pick up where we left off last time we're going to continue our exposition of a particle on a sphere which is going to be our entree into the theory of angular momentum that later on we're going to use when we talk about Real and today than particle on a sphere and angular remember that we had our separated equations we had by dividing by the product of state of status and Fireside we ended up with a sum of several terms 1 of the terms only has 5.

- Angular momentum in spherical coordinates Peter Haggstrom www.gotohaggstrom.com mathsatbondibeach@gmail.com December 6, 2015 1 Introduction Angular momentum is a deep property and in courses on quantum mechanics a lot of time is devoted to commutator relationships and spherical harmonics. However, man
- For a rigid body rotating on an axis (e.g., the Earth spinning), the angular momentum is the product of the moment of inertia (I) and the angular velocity (w): L rot = I*w, and for a rigid, spherical body, I = 0.4MR 2
- A solid sphere has an angular velocity of omega (w) and a rotational kinetic energy of K. Write an expression for the sphere's angular momentum in terms of omega (w) and K. angular-momentum
- So the total angular momentum for the whole sphere is M v r and this is the orbital angular momentum that your teacher was referring to. You take the momentum of the centre of mass M v and multiply it by the distance between the line of motion and the point

In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of linear momentum. It is an important quantity in physics because it is a conserved quantity —the total angular momentum of a closed system remains constant We now proceed to calculate the angular momentum operators in spherical coordinates. The first step is to write the in spherical coordinates. We use the chain rule and the above transformation from Cartesian to spherical ** The angular momentum of a sphere is **. Embed a running copy of this simulatio

The angular momentum of an infinitesimal element with respect to the spin axis of the vortex is the areal density ρ times the velocity times the distance s of the element from the spin axis. The distance s of a point at angle θ from the spin axis is Rsin (θ) A sphere that has the Cartesian equation x 2 + y 2 + z 2 = c 2 has the simple equation r = c in spherical coordinates. Two important partial differential equations that arise in many physical problems, Laplace's equation and the Helmholtz equation, allow a separation of variables in spherical coordinates. The angular portions of the solutions to such equations take the form of spherical. * This is a demonstration of the conservation of angular momentum using a Hoberman sphere, a plastic sphere frame that can be contracted by pulling on a string*.. There is a vector that contains direction of the axis of rotation. The magnitude of angular momentum of the particle that travels around the sphere can be defined as: (1) J = p

* Rolling, Torque, and Angular Momentum Rolling Motion: • A motion that is a combination of rotational and translational motion, e*.g. a wheel rolling down the road. • Will only consider rolling with out slipping. For a disk or sphere rolling along a horizontal surface, the motion can be considered in two ways: I. Combination of rotational and translational motion: • Center of mass moves in. The polarization state of a light beam is related to its spin angular momentum and can be represented on the Poincaré sphere. We propose a sphere for light beams in analogous orbital angular momentum states. Using the Poincaré-sphere equivalent, we interpret the rotational frequency shift for light The angular momentum of a sphere is given by L = (−5.00t 3) + (6.10 − 1.80t 2) + (6.95t), where L has units of kg · m 2 /s when t is in seconds

the moment of inertia tensor is. (2) (3) (4) which is diagonal, and so it is in principal axis form. Furthermore, because of the symmetry of the sphere, each principal moment is the same, so the moment of inertia of the sphere taken about any diameter is . Moment of Inertia, Moment of Inertia--Spherical Shell Gyroscopes used in guidance systems to indicate directions in space must have an angular momentum that does not change in direction. When placed in the vehicle, they are put in a compartment that is separated from the main fuselage, such that changes in the orientation of the fuselage does not affect the orientation of the gyroscope The structure of Equation 1.6.2 suggests that this angular-momentum operator is given by. ^ Lz = − iℏ ∂ ∂ϕ. This result will follow from a more general derivation in the following Section. The Schrödinger equation (Equation 1.6.2) can now be written more compactly as. ψ′′ (ϕ) + m2ψ(ϕ) = 0. where. m2 ≡ 2IE / ℏ2 * Spin Angular Momentum introduces an entirely new note type: Angle notes*. These notes change the view of the camera. +/-45' notes rotate about the current view, while +/-90' notes rotate to a different view. They also give +10 combo, much like silver notes Quantum Principles: Particle on a **Sphere**, **Angular** **Momentum**. If playback doesn't begin shortly, try restarting your device. Videos you watch may be added to the TV's watch history and influence TV.

Scattering of Electromagnetic Waves With Orbital Angular Momentum on Metallic Sphere Abstract: In this letter, theory concerning scattering of electromagnetic (EM) waves with orbital angular momentum (OAM) is presented, in which a metallic sphere illuminated by OAM waves is adopted to simplify the study The angular momentum due to the earth's rotation is ~~7.2times 10^33\ Kg\ m^2s^-1 (this value is with respect to a co-moving observer) We can estimate the angular momentum due to the earth's rotation by approximating the earth by a uniform sphere of mass M= 6.0times 10^24\ Kg and radius R = 6.4 times 10^6\ m The moment of inertia of a uniform solid sphere about any axis passing. Hoberman spheres, invented by Chuck Hoberman, are beautifully engineered icosidodecahedrons that are fun to expand and contract. This one can be contracted by pulling a string. As it turns out, a Hoberman sphere is also an excellent tool for demonstrating the conservation of angular momentum.In the video above, Utah State University's Professor Boyd F. Edwards spins the expanded sphere, then. let's talk a little bit about the conservation of angular momentum and this is going to be really useful because it explains diverse phenomena in the universe from Y and ice skaters angular speed goes up when they tuck their arms or their legs in all the way to when you have something orbiting around a star and the closer and closer it spirals in it seems like it's rotational velocities. We analyzed the spin and orbital angular momenta of magnons in the Walker modes and photons in the WGMs, both of which being supported by a ferromagnetic sphere. We then predicted that in the Brillouin light scattering within the ferromagnetic sphere the orbital angular momenta are exchanged between the photons and the magnons in such a way that the total orbital angular momentum is conserved.

- The concept might not seem too interesting at first, but in combination with the law of the conservation of angular momentum, it can be used to describe many fascinating physical phenomena and predict motion in a wide range of situations. Definition of Moment of Inertia. The moment of inertia for an object describes its resistance to angular acceleration, accounting for the distribution of.
- A higher-order Poincaré sphere and Stokes parameter representation of the higher-order states of polarization of vector vortex beams that includes radial and azimuthal polarized cylindrical vector beams is presented. The higher-order Poincaré sphere is constructed by naturally extending the Jones vector basis of plane wave polarization in terms of optical spin angular momentum to the total.
- What is the angular momentum of the sphere? 01:53. Video Transcript. A solid metal sphere is rotating with an angular speed of 12.6 radians per second. It has a moment of inertia of 0.0200 kilograms meters squared. What is the angular momentum of the sphere? Let's say that this here is our sphere, and in our problem statement, we're told that it's rotating with a certain angular speed.
- e spin, it is independent of the onn angular momentum. The net spin of the two onta.
- Practice: When solid sphere 4 m in diameter spins around its central axis at 120 RPM, it has 1,000 kg m 2 / s in angular momentum. Calculate the sphere's mass. Practice: A composite disc is built from a solid disc and a concentric, thick-walled hoop, as shown below. The inner disc has mass 4 kg and radius 2 m. The outer disc (thick-walled) has mass 5 kg, inner radius 2 m, and outer radius 3.

Angular Momentum: Moment of Inertia--Sphere For a solid sphere with radius R, mass M, and density, (1) the moment of inertia tensor is (2) (3) (4) which is diagonal, and so it is in principal axis form. Furthermore, because of the symmetry of the sphere, each principal moment is the same, so the moment of inertia of the sphere taken about any diameter is . Moment of Inertia, Moment of Inertia. The angular momentum of a sphere is . Embed a running copy of this simulation. Net angular momentum at time ti = Net angular momentum at later time tf. If the component of the net external torque on a system along a certain axis is zero, the component of the angular momentum of the system along that axis cannot change, no matter what changes take place within the system. This conservation law holds not only within the frame of Newton's mechanics but also for. ** AP Physics Practice Test: Rotation, Angular Momentum ©2011, Richard White www**.crashwhite.com 3. 2 A solid sphere of mass m is fastened to another sphere of mass 2m by a thin rod with a length of 3x.The spheres have negligible size, and the rod has negligible mass

Hoberman sphere conservation of angular momemntum. Play. 0:00. 0:00. Settings. Fullscreen. 131 comments. share. save. hide. report. 96% Upvoted. This thread is archived. New comments cannot be posted and votes cannot be cast . Sort by. best. View discussions in 1 other community. level 1. 850 points · 2 years ago. When ballerinas spin they extend their arms then pull them in, I thought the. Conservation of Angular Momentum - Hoberman Sphere . Condition: Good . Principle: Conservation of Angular Momentum . Area of Study: Mechanics . Disclaimer. Equipment: Hoberman Sphere Mobile. Procedure: See also 1Q40.22 in Mechanics. Hang the Hoberman sphere mobile up by the supplied rope in LR1 or LR2. In LR 70 the mobile will have to be suspended from a cross arm on one of the tables. Spin.

- Angular momentum relates to how much an object is rotating. An object has a constant angular momentum when it is neither speeding up nor slowing down. The angular momentum of an object depends on the distribution of the mass of the object. The moment of inertia is a value that describes the distribution. It can be found by integrating over the mass of all parts of the object and their.
- Angular Momentum of a Single Particle. shows a particle at a position . with linear momentum . with respect to the origin. Even if the particle is not rotating about the origin, we can still define an angular momentum in terms of the position vector and the linear momentum
- The angular momentum [latex] \overset{\to }{l} [/latex] of a particle is defined as the cross-product of [latex] \overset{\to }{r} [/latex] and [latex] Which has greater angular momentum: a solid sphere of mass m rotating at a constant angular frequency [latex] {\omega }_{0} [/latex] about the z-axis, or a solid cylinder of same mass and rotation rate about the z-axis? Show Solution. Visit.
- This equation is an analog to the definition of linear momentum as p = mv.Units for linear momentum are kg ⋅ m/s while units for angular momentum are kg ⋅ m 2 /s. As we would expect, an object that has a large moment of inertia I, such as Earth, has a very large angular momentum.An object that has a large angular velocity ω, such as a centrifuge, also has a rather large angular momentum
- Quantum Principles: Particle on a Sphere, Angular Momentum Home. Chem 131A. Lec 11. Quantum Principles: Particle on a Sphere, Angular Momentum (English) UCI Chem 131A Quantum Principles (Winter 2013) Lec 11. Quantum Principles -- Particle on a Sphere, Angular Momentum --.
- ally zero, which illustrates a limitation of this form for inﬁnite solenoids [6]. 3. A Appendix: Rotating Spherical Shell of Charge The case of a spherical shell.

Angular momentum carried by light • Cartesian coordinates on the Poincaré Sphere are normalized Stokes parameters: P 1 /P 0, P 2 /P 0, P 3 /P 0 • With some trigonometry, one can see that a state of arbitrary polarization is represented by a point on the Poincaré Sphere of unit radius: • Partially polarized light ⇒R<1 • R ≡degree of polarization 222 12 3 0 1 PPP R P ++ == 20. Then we obtain the rate at which angular momentum is radiated away by the shell, and the total angular momentum contained in the electromagnetic field. With these results we demonstrate explicitly that the field angular momentum is lost in part to radiation and in part to the self-torque; angular momentum balance is thereby established. Finally, we examine the angular motion of the sphere. Harmonics, Angular Momentum We can now extend the Rigid Rotor problem to a rotation in 3D, corre-sponding to motion on the surface of a sphere of radius R. The Hamiltonian operator in this case is derived from the Laplacian in spherical polar coordi-nates given as ∇2 = ∂ 2 ∂x 2 + ∂ ∂y + ∂2 ∂z = ∂2 ∂r 2 + 2 r ∂ ∂r + 1 r 1 sin θ ∂2 ∂φ + 1 sinθ ∂ ∂θ sinθ.

- between the two spheres, and the total EM angular momentum of the system is readily found to be 0. = 8 9 (. 0. . 3. . 2. Ω. 0)(. 2. −. 02) . (3) It is thus seen that, in the limit when . 0 →0, the two angular momenta, calculated with and without the second sphere, are identical. In one case, the particle.
- David explains how a mass can have angular momentum even if it is traveling along a straight line. Then David shows how to solve the conservation of angular momentum problem where a ball hits a rod which can rotate. If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter, please make sure that the domains *.kastatic.
- 本页面最后编辑于2020年4月24日 (星期五) 16:27。 除非另有声明，本网站内容采用知识共享署名-非商业性使用-相同方式共享授权。; 隐私政策; 关于Tone Sphere 中文维
- A hollow sphere and a hollow cylinder of the same radius and mass roll up an incline without slipping and have the same initial center of mass velocity. Which object reaches a greater height before stopping? 11.2 Angular Momentum. Can you assign an angular momentum to a particle without first defining a reference point? For a particle traveling in a straight line, are there any points about.
- Angular momentum is conserved, so calculate the initial angular momentum of the sphere as measured from the pivot, and set that equal to the angular momentum of the rod immediately after the collision. Energy is conserved, so calculate the initial kinetic energy of the sphere immediately before the collision, and set that equal to the kinetic energy of the rod immediately after the collision.
- ary Questions 1. With the torque being constant, how is the moment of inertia related to the angular acceleration? 2. If the angular momentum stays constant, how is the moment of inertia related to the angular velocity? 3. Take a solid cylinder and a solid sphere both with same radii and mass.

Angular Momentum Formula Questions: 1) A DVD disc has a radius of 0.0600 m, and a mass of 0.0200 kg.The moment of inertia of a solid disc is , where M is the mass of the disc, and R is the radius. When a DVD in a certain machine starts playing, it has an angular velocity of 160.0 radians/s.What is the angular momentum of this disc Contributors and Attributions; Broadly speaking, a classical extended object (e.g., the Earth) can possess two different types of angular momentum.The first type is due to the rotation of the object's center of mass about some fixed external point (e.g., the Sun)—this is generally known as orbital angular momentum.The second type is due to the object's internal motion—this is generally. Angular Momentum of a Single Particle. Figure shows a particle at a position [latex]\mathbf{\overset{\to }{r}}[/latex] with linear momentum [latex]\mathbf{\overset{\to }{p}}=m\mathbf{\overset{\to }{v}}[/latex] with respect to the origin. Even if the particle is not rotating about the origin, we can still define an angular momentum in terms of the position vector and the linear momentum A higher-order Poincar\'e sphere and Stokes parameter representation of the higher-order states of polarization of vector vortex beams that includes radial and azimuthal polarized cylindrical vector beams is presented. The higher-order Poincar\'e sphere is constructed by naturally extending the Jones vector basis of plane wave polarization in terms of optical spin angular momentum to the total. In quantum mechanics, angular momentum is a vector operator of which the three components have well-defined commutation relations. The endpoint of the blue arrow does not cover the full surface of the sphere, but can only lie on the intersection of the sphere and certain quantized cones, because the projection m of j on the z-axis is quantized. The discrete quantum number m is integral or.

430 OPTICS LETTERS / Vol. 24, No. 7 / April 1, 1999 Poincare-sphere equivalent for light beams containing orbital´ angular momentum M. J. Padgett and J. Courtial School of Physics & Astronomy. Angular momentum balance is examined in the context of the electrodynamics of a spinning charged sphere, which is allowed to possess any variable angular velocity. We calculate the electric and magnetic fields of the (hollow) sphere, and express them as expansions in powers of τ/t c ≪ 1, the ratio of the light-travel time τ across the sphere and the characteristic time scale t<SUB>c</SUB. So now the excesses passing producen at the center of the orbit and angular momentum warned This excess is we went off with her, she off the art which is I m into the distance between the art and son. It's what that just say this distances be these the radius off this orbit. So movement up in a job that is MD square into the angular velocity Must is six into 10 to the power 24 and the radiance. The polarization state of a light beam is related to its spin angular momentum and can be represented on the Poincaré sphere. We propose a sphere for light beams in analogous orbital angular momentum states. Using the Poincaré-sphere equivalent, we interpret the rotational frequency shift for light beams with orbital angular momentum [Phys.??Rev.??Lett.??80, 3217 (1998)] as a dynamically.

- Angular momentum in terms of Linear momentum can be written as \(L=r\times p\) where, r = length vector. p = linear momentum. The unit for Angular momentum is given as kilogram meter square per second (kg m 2 /s). Angular Momentum formula is made use of in computing the angular momentum of the particle and also to find the parameters associated to it. Angular Momentum Numericals. Problem 1: A.
- How would the angular momentum of the sphere at the bottom of the plane change if the coefficient of static friction between the sphere and the plane were increased? answer choices . It would decrease because the force of friction would be greater. It would decrease because the net torque would be larger. It would increase because the angular acceleration would increase. It would increase.
- The volunteer holds it horizontally and spins himself, he will rotate in opposite direction to conserve angular momentum; The demonstrations illustrate the vector value of the law of conservation of angular momentum. Top . M.14(3) - Tippe-Top . The top consists of a hemisphere and a control stem. When it is spun on a flat surface, it spins on its spherical end for only a few seconds and then.

What is its angular momentum? Question. help_outline. Image Transcriptionclose. A hollow, iron sphere has a radius R = 0.8 m and mass M = 8 kg. IT rotates at a constant àngular speed of 1.2 rad/s about an axis veritcal and passing through-its center. What is its angular momentum? fullscreen . check_circle Expert Answer. Want to see the step-by-step answer? See Answer. Check out a sample Q&A. Conservation of Angular Momentum: Hoberman sphere. Enlighten Science. October 4, 2019 · Demo of conservation of angular momentum with the Hoberman sphere. Related Videos. 3:34. A sphere is released on a smooth inclined plane from the top. When it moves down, its angular momentum is. A. conserved about every point. B. conserved about the point of contact only. C. conserved about the centre of the sphere only . D. conserved about any point on a line parallel to the inclined plane and passing through the centre of the ball. Medium. Answer. Correct option is . D. Calculate the angular momentum for a rotating disk, sphere, and rod. (a) A uniform disk of mass 16 kg, thickness 0.5 m, and radius 0.6 m is located at the origin, oriented with its axis along the y axis

The Angular Momentum using radius formula is defined as the rotational equivalent of linear momentum. It is an important quantity in physics because it is a conserved quantity is calculated using angular_momentum = Mass * Velocity * Radius.To calculate Angular Momentum using radius, you need Mass (m), Velocity (v) and Radius (r).With our tool, you need to enter the respective value for Mass. We present a new optical effect that exchanges angular momentum between light and matter. The matter consists of an optically thick spherical, rigid agglomerate of magneto-optical scatterers placed in a homogeneous magnetic field. The light comes from an unpolarized, coherent central light source. The photon Hall effect induces a spiraling Poynting vector, both inside and outside the medium As the angular momentum decreases to zero, the ergosphere disappears as the black hole becomes a Schwarzchild black hole. Another cool thing about rotating black holes is the existence of a ring singularity (ringularity). In a Schwarzchild black hole the center of the black hole is a point of infinite density. For a rotating black hole, it is no longer a point, but a circle of infinite density. We use astigmatic transformations to characterize two-dimensional superpositions of orbital-angular-momentum states in laser beams. We propose two methods for doing this, both relying only on astigmatic transformations, viewed as rotations on the Poincar\'e sphere, followed by imaging. These methods can be used as a tomographic tool for communication protocols based on optical vortices Tone **Sphere** is a mobile music-rhythm game developed by Sta Kousin of Bit192 Labs. The current version (as of February 2021) is 1.9.4, with the latest update adding the song Glitched World to Darksphere.; Available for iOS and Android devices [edit | edit source]. The base game, with the episodes Press/Start, Solarsphere, Darksphere and Darksphere XXXL, is completely free

Numerical Problems on Rolling motion, Torque, and Angular Momentum (worksheet with medium & Hard problems) 10 ) A homogeneous sphere starts from rest at the upper end of the track shown below and rolls without slipping until it rolls off the right-hand end. If H = 60 m and h = 20 m and the track is horizontal at the right-hand end, determine the distance to the right of point A at which. The angular momentum of sphere : L = I ω = (0.032)(4) = 0.128 kg m 2 /s . Read : Resonance of sound wave - problems and solutions. 4. A 1-kg particle rotates at a constant angular speed of 2 rad/s. What is the angular speed if the radius of circle is 1 0 cm. Known : Mass of object (m) = 1 k g. The radius of circle (r) = 10 cm = 10/100 = 0.1 m. The angular speed (ω) = 2 rad/ s. Wanted.

Chapter 6 Orbital Angular Momentum and Angular Functions on the Sphere 269 1. Rotational Symmetry of a Simple Physical System 270 2. Scalar Product of State Vectors 271 3. Unitarity of the Orbital Rotation Operator 272 4. A (Dense) Subspace of %(S) 273 5. Only Integral Values of / can occur in the Quantization of Spatial (Orbital) Angular Momentum 274 6. Transformations of the Solid Harmonics. The number of degrees of freedom for a sphere is a bit uncertain. It could be three for rotations about three orthogonal axes. It could be just one for a charged sphere. The angular momentum according the Bohr-Mottelson Rule is L = h(I(I+1)) ½. Thus h(I(I+1)) ½ = 9.274((½h) and therefore ( I(I+1)) ½ = ½(9.274) = 4.637 which means (I(I+1)) = 21.5 ≅ 4(5) Thus, this suggests that the. The basic definition of angular momentum is the cross product of a position vector with linear momentum, r x p. r is measured from a hypothetical axis of rotation, and p in Newtonian Physics is mv. Here hypothetical means that more than one axis..

If the Democrats and their media allies were absolutely confident in the validity of the election results you would think they would welcome an audit (not a mere recount) of the ballots in Maricopa County, Arizona.. sphere (Figure 4, right). These two simple models capture the essence of the origin of electron orbital angular momentum and to electron spin angular momentum. We choose two definite physical models to visualize (1) orbital angular momentum in terms of an electron of mass me travelling in a circular Bohr orbit or radius r with angular velocity v, and (2) spin angular momentum in terms of a.

modeling it as a uniform sphere. 24 m E u5.97 10 kg 6 R E u6.38 10 m Orbital radius r u1.50 10 m11 Period of rotation P rot = 24 h 86,400 s Period of revolution P rev = 1 y 3.156 10 s u7 Identify: zz LI Z Set Up: For a particle of mass m moving in a circular path at a distance r from the axis, I mr2 and vr Z. For a uniform sphere of mass M and radius R and an axis through its center, 2 5 I MR. Rigid Rotor Summary Up: The Rigid Rotor Previous: Energy Angular Momentum. Rotation is intimately connected with ANGULAR MOMENTUM and the relationship between the Hamiltonian, and the square of the rotation operator means that wavefunctions that satisify one, satisfy the other; moreover, the observable values of both operators are related.. Thus, in a similar manner to the angular momentum. Left: Body angular momentum sphere and four paths in the triaxial case. Right: Body angular rate components and their magnitude in the case (3). Figure 7.6. Left: Time profiles of the angular momentum coordinates and their magnitude in the case (3). Right: Four-period herpolhode in the case (3). The Jacobi elliptic functions are periodic functions defined on the unit ellipse (x/a) 2 +y 2 = 1. Angular momentum balance is examined in the context of the electrodynamics of a spinning charged sphere, which is allowed to possess any variable angular velocity. We calculate the electric and magnetic fields of the (hollow) sphere, and express them as expansions in powers of τ / t c ≪ 1, the ratio of the light-travel time τ across the sphere and the characteristic time scale t c of. C. the sphere reaches the bottom ﬁrst because it picks up more rotational energy D. they reach the bottom together E. none of the above are true ans: E 10. A hoop, a uniform disk, and a uniform sphere, all with the same mass and outer radius, start with the same speed and roll without sliding up identical inclines. Rank the objects according to how high they go, least to greatest. A. hoop.

Which has greater angular momentum: a solid sphere of mass m rotating at a constant angular frequency ω 0 ω 0 about the z-axis, or a solid cylinder of same mass and rotation rate about the z-axis? Interactive. Visit the University of Colorado's Interactive Simulation of Angular Momentum to learn more about angular momentum. Previous Next. Order a print copy. As an Amazon Associate we earn. angular momentum, we will conclude that, with care, the form L(M) is the most reliable for computation of ﬁeld angular momentum, that form L(F) should not be used, and that the form L(P) is valid only for ﬁelds that fall oﬀ suﬃciently quickly at large distances (which is not the case for an inﬁnite solenoid). 2.1 Inﬁnite Solenoid Consider an inﬁnite solenoid of radius R about the. 3/34 1 Introduction 1.1 Subject In order to unify gravity and electromagnetic force, we introduced 1-sphere in extra dimensional space. In order to derive two-valuedness and angular momentum of spin-1/2, we introduce 3-spher