Two large metal plates (each of area A) are held a small distance d apart. Suppose we put a charge Q on each plate; what is the electrostatic pressure on the plates?
> A sphere of radius R, centered at the origin, carries charge density where k is a constant, and r , θ are the usual spherical coordinates. Find the approximate potential for points on the z axis, far from the sphere.
> Charge density σ(φ) = a sin 5φ (where a is a constant) is glued over the surface of an infinite cylinder of radius R (Fig. 3.25). Find the potential inside and outside the cylinder. [Use your result from Pr
> Find the potential outside an infinitely long metal pipe, of radius R, placed at right angles to an otherwise uniform electric field E0. Find the surface charge induced on the pipe. [Use your result from Prob. 3.24.]
> Solve Laplace’s equation by separation of variables in cylindrical coordinates, assuming there is no dependence on z (cylindrical symmetry). [Make sure you find all solutions to the radial equation; in particular, your result must accommodate the case of
> A spherical shell of radius R carries a uniform surface charge σ0 on the “northern” hemisphere and a uniform surface charge σ0 on the “southern” hemisphere. Find the potential inside and outside the sphere, calculating the coefficients explicitly up to A6
> In Prob. 2.25, you found the potential on the axis of a uniformly charged disk: / (a) Use this, together with the fact that Pl (1) 1, to evaluate the first three terms in the expansion (Eq. 3.72) for the potential of the disk at points off the axis, assu
> Suppose that f is a function of two variables (y and z) only. Show that the gradient / transforms as a vector under rotations, Eq. 1.29. and the analogous formula for /. We know that / And / “solve” these equat
> Find the potential outside a charged metal sphere (charge Q, radius R) placed in an otherwise uniform electric field E0. Explain clearly where you are setting the zero of potential.
> Suppose the potential V0(θ) at the surface of a sphere is specified, and there is no charge inside or outside the sphere. Show that the charge density on the sphere is given by / where /
> In one sentence, justify Earnshaw’s Theorem: A charged particle cannot be held in a stable equilibrium by electrostatic forces alone. As an example, consider the cubical arrangement of fixed charges in Fig. 3.4. It looks,
> The potential at the surface of a sphere (radius R) is given by V0 = k cos 3θ, where k is a constant. Find the potential inside and outside the sphere, as well as the surface charge density σ(θ) on the sphere. (Assume there’s no charge inside or outside
> (a) Suppose the potential is a constant V0 over the surface of the sphere. Use the results of Ex. 3.6 and Ex. 3.7 to find the potential inside and outside the sphere. (Of course, you know the answers in advance—this is just a consistency check on the meth
> Derive P3(x) from the Rodrigues formula, and check that P3(cos θ) satisfies the angular equation (3.60) for l=3. Check that P3 and P1 are orthogonal by explicit integration.
> A cubical box (sides of length a) consists of five metal plates, which are welded together and grounded (Fig. 3.23). The top is made of a separate sheet of metal, insulated from the others, and held at a constant potential V0. Find the pot
> A rectangular pipe, running parallel to the z-axis (from −∞ to +∞), has three grounded metal sides, at y = 0, y = a, and x = 0. The fourth side, at x = b, is maintained at a specified potential V0(y). (a) Develop a general formula for the potential inside
> For the infinite slot (Ex. 3.3), determine the charge density σ(y) on the strip at x = 0, assuming it is a conductor at constant potential V0.
> Find the potential in the infinite slot of Ex. 3.3 if the boundary at x = 0 consists of two metal strips: one, from y = 0 to y = a/2, is held at a constant potential V0, and the other, from y = a/2 to y = a, is at potential −V0.
> Let r be the separation vector from a fixed point (x r, yr, zr) to the point (x, y, z), and let r be its length. Show that (a) ∇(r2) = 2r. (b) ∇(1/r) = −rˆ/r2. (c) What is the general formula for ∇(rn)?
> Two long, straight copper pipes, each of radius R, are held a distance 2d apart. One is at potential V0, the other at V0 (Fig. 3.16). Find the potential everywhere. [Hint: Exploit the result of Prob. 2.52.]
> Two semi-infinite grounded conducting planes meet at right angles. In the region between them, there is a point charge q, situated as shown in Fig. 3.15. Set up the image configuration, and calculate the potential in this re
> A uniform line charge λ is placed on an infinite straight wire, a distance d above a grounded conducting plane. (Let’s say the wire runs parallel to the x -axis and directly above it, and the conducting plane is the xy plane.) (a) Find the potential in th
> Find the average potential over a spherical surface of radius R due to a point charge q located inside (same as above, in other words, only with z < R). (In this case, of course, Laplace’s equation does not hold within the sphere.) Show that, in general,
> Suppose the electric field in some region is found to be / in spherical coordinates (k is some constant). (a) Find the charge density ρ. (b) Find the total charge contained in a sphere of radius R, centered at the origin. (Do it two different ways.)
> Use your result in Prob. 2.7 to find the field inside and outside a solid sphere of radius R that carries a uniform volume charge density ρ. Express your answers in terms of the total charge of the sphere, q. Draw a graph of (E( as a function of the distan
> Find the electric field a distance z from the center of a spherical surface of radius R (Fig. 2.11) that carries a uniform charge density σ. Treat the case z R (outside). Express your answers in terms of the total charge q on t
> What is the minimum-energy configuration for a system of N equal point charges placed on or inside a circle of radius R?17 Because the charge on a conductor goes to the surface, you might think the N charges would arrange themselves (uniformly) around th
> A point charge q is at the center of an uncharged spherical conducting shell, of inner radius a and outer radius b. Question: How much work would it take to move the charge out to infinity (through a tiny hole drilled in the shell)? [Answer: /
> Find the electric field a distance z above the center of a flat circular disk of radius R (Fig. 2.10) that carries a uniform surface charge σ. What does your formula give in the limit R → â
> The height of a certain hill (in feet) is given by h(x, y) = 10(2xy − 3x 2 − 4y2 − 18x + 28y + 12), where y is the distance (in miles) north, x the distance east of South Hadley. (a) Where is the top of the hill located? (b) How high is the hill? (c) How
> Prove or disprove (with a counterexample) the following Theorem: Suppose a conductor carrying a net charge Q, when placed in an external electric field Ee, experiences a force F; if the external field is now reversed (Ee → −Ee), the force also reverses (
> (a) Consider an equilateral triangle, inscribed in a circle of radius a, with a point charge q at each vertex. The electric field is zero (obviously) at the center, but (surprisingly) there are three other points inside the triangle where the field is zero
> We know that the charge on a conductor goes to the surface, but just how it distributes itself there is not easy to determine. One famous example in which the surface charge density can be calculated explicitly is the ellipsoid: In this case15 / where Q
> All of electrostatics follows from the 1/r2 character of Coulomb’s law, together with the principle of superposition. An analogous theory can therefore be constructed for Newton’s law of universal gravitation. What is the gravitational energy of a sphere
> Suppose an electric field E(x, y, z) has the form Ex = ax, Ey = 0, Ez = 0 where a is a constant. What is the charge density? How do you account for the fact that the field points in a particular direction, when the charge density is uniform? [This is a mor
> Imagine that new and extraordinarily precise measurements have revealed an error in Coulomb’s law. The actual force of interaction between two point charges is found to be where λ is a new constant of nature (it has dimensi
> In a vacuum diode, electrons are “boiled” off a hot cathode, at potential zero, and accelerated across a gap to the anode, which is held at positive potential V0. The cloud of moving electrons within the gap (called sp
> Two infinitely long wires running parallel to the x axis carry uniform charge densities +λ and −λ (Fig. 2.54). (a) Find the potential at any point (x, y, z), using the origin as your reference. (
> Find the potential on the rim of a uniformly charged disk (radius R, charge density σ ). [Hint: First show that / for some dimensionless number k, which you can express as an integral. Then evaluate k analytically, if you can, or by computer.]
> The electric potential of some configuration is given by the expression where A and λ are constants. Find the electric field E(r), the charge density ρ(r), and the total charge Q. /
> Find the gradients of the following functions: (a) f (x, y, z) = x 2 + y3 + z4. (b) f (x, y, z) = x 2 y3z4. (c) f (x, y, z) = ex sin(y) ln(z).
> Find the electric field a distance z above the center of a circular loop of radius r (Fig. 2.9) that carries a uniform line charge λ.
> A sphere of radius R carries a charge density ρ(r) kr (where k is a constant). Find the energy of the configuration. Check your answer by calculating it in at least two different ways.
> An inverted hemispherical bowl of radius R carries a uniform surface charge density σ. Find the potential difference between the “north pole” and the center.
> Find the net force that the southern hemisphere of a uniformly charged solid sphere exerts on the northern hemisphere. Express your answer in terms of the radius R and the total charge Q. /
> If the electric field in some region is given (in spherical coordinates) by the expression for some constant k, what is the charge density? /
> Find the electric field at a height z above the center of a square sheet (side a) carrying a uniform surface charge σ. Check your result for the limiting cases a →∞ and z ((a.
> Suppose the plates of a parallel-plate capacitor move closer together by an infinitesimal distance ε as a result of their mutual attraction. (a) Use Eq. 2.52 to express the work done by electrostatic forces, in terms of the field E , and the area of the pl
> Find the capacitance per unit length of two coaxial metal cylindrical tubes, of radii a and b (Fig. 2.53).
> A metal sphere of radius R carries a total charge Q. What is the force of repulsion between the “northern” hemisphere and the “southern” hemisphere?
> (a) How do the components of a vector5 transform under a translation of coordinates / (b) How do the components of a vector transform under an inversion of coordinates / (c) How do the components of a cross product (Eq. 1.13) transform under inversio
> Using the definitions in Eqs. 1.1 and 1.4, and appropriate diagrams, show that the dot product and cross product are distributive, a) when the three vectors are coplanar;! b) in the general case.
> This exercise tests your understanding of all aspects of accounting learned and tests your ability to use an accounting equation spreadsheet for recording the transactions of a limited company. Information about the business The opening statement of fin
> Set out below is a summary of the accounting records of Titan Ltd at 31 December Year 1: The summary of the accounting records includes all transactions which have been entered in the ledger accounts up to 31 December, but investigation reveals further a
> Explain why the following item is reported as a contingent liability and not as a provision. ‘Under the Value Added Incentive Scheme, which is currently under review by the Board, further amounts may be payable to certain directors and employees. The amo
> The following information is taken from the notes of the statement of financial position (balance sheet) of a listed company. Write a short explanation that is suitable for a private shareholder who does not have specialist accountancy training. The loan
> The following information is taken from the annual report of a major listed company. Write a short explanation that is suitable for a private shareholder who does not have specialist accountancy training. Note to the accounts: Non-current (long-term) li
> Explain why each of the following is reported as a contingent liability but not recognized as a provision in the statement of financial position (balance sheet). (a) Some leasehold properties which the group no longer requires have been sublet to third p
> Explain why each of the following is recognized as a provision in the statement of financial position (balance sheet) of a telecommunications company: (a) On 15 December Year 2, the Group announced a major redundancy programmed. Provision has been made a
> The accountant of Blue Ltd discovered the following documents in a file marked ‘not yet dealt with’ at the company’s year end of 31 December Year Explain the accounting treatment required for each. (a) A supplier’s invoice for £300 was received on 21 De
> The statement of financial position (balance sheet) of a furniture retail company includes in current liabilities the item: ‘Deposits received from customers’ Explain why this item is a current liability.
> The statement of financial position (balance sheet) of an electricity company includes in current liabilities the item: ‘Prepayments by electricity consumers’ Explain why this item is a current liability.
> The following transactions relate to Computer Assembler Company during the month of May. Required: (a) Calculate the profit on sale. (b) Explain the effect of each transaction on the accounting equation. (c) Make entries in a spreadsheet summarizing t
> Discuss the problems of comparability of revenue recognition based on the following information: In the UK film and television industry, a range of accounting practices has been used in recognizing revenue. Three examples are as follows: Company A: This
> It is the policy of Hellebore Ltd to make provisions for doubtful debts at a rate of 15 per cent per annum on all debtor balances at the end of the year, after deducting any known bad debts on the same date. The following table sets out the total receiva
> Canyon Ltd uses FIFO stock valuation. Canyon has recently been acquired by a US parent company. The directors in the US wish to know what the profits of the past 5 years would have been if LIFO stock valuation had been applied. Relevant data are provided
> A fire destroyed a company’s detailed records of inventory (stock) and much of the merchandise held in inventory (stock). The company accountant was able to discover that stock at the beginning of the period was £60,000, purchases up to the date of the f
> It is the policy of Seaton Ltd to make provision for doubtful debts at a rate of 10% per annum on all debtor balances at the end of the year, after deducting any known bad debts at the same date. The following table sets out the total receivables (debtor
> A fire destroyed a company’s detailed records of inventory (stock) and much of the merchandise held in inventory (stock). The company accountant was able to discover that stock at the beginning of the period was £60,000, purchases up to the date of the f
> The Machine Grinding Company commenced business on 1 January Year 1 with capital of £80,000 contributed by the owner. The company immediately paid cash for a grinding machine costing £80,000. It was estimated to have a useful life of 6 years and at the e
> A food processing company purchased, on 1 January Year 1, a canning machine. The machine costs £24,000 and was estimated to have a 5-year life with a residual value of £2,000. Required: (a) Prepare a table of depreciation charges and net book value ove
> The Machine Grinding Company sold the machine for £40,000 at the start of Year 4 and replaced it with a new machine costing £100,000. The new machine had an estimated useful life of 8 years and an estimated residual value of £10,000 at the end of that pe
> Brian Grant set up a business selling quality electrical goods on 1 May 2001. The following represents the business transactions of the first month. 1. Brian paid £250,000 to the business bank account. 2. He obtained a 10-year business loa
> The following list of transactions relates to a television repair business, Peter (Television) Company, during July, which is its first month of business. Required: (a) Analyze the above transactions to show the two aspects of each transaction. (b) Prep
> Report and Accounts of Craigielaw plc 1. The statement of financial position (balance sheet) contains broad headings representing categories of assets and liabilities, but greater detail is found in the notes to the accounts. Follow the information in th
> 1. Look at the Property, Plant and Equipment note to the statement of financial position (balance sheet). What are the headings to the columns of figures? Explain how each heading passes the test of definition of an asset. 2. Look at the group statement
> 1. What type of business entity is Craigielaw? 2. What is the company’s business activity? 3. How many shareholders are there in Craigielaw? How many shares are there in Craigielaw? Do all shareholders own approximately an equal number of shares in the c
> 1. Scrutinise all sections of the Report and Accounts of Craigielaw plc and note all instances where accounting ratios have been mentioned. 2. What are the most frequently mentioned ratios in the Report and Accounts of Craigielaw plc? 3. Are there ratios
> 1. The ‘Statement of comprehensive income’ and the ‘Statement of changes in equity’ are intended to help the reader understand how the ownership interest has changed over the account
> 1. General types of non-current (long-term) liabilities (from Statement of financial Position. 2. Characteristics of loans: secured or unsecured? Information on Loans is contained in notes 21 and 20 on pages 33 and 34. You should be able to complete each
> 1. What is the nature and significance of individual current liability items in the financial statements? Complete the following table and write a short comment. 2. What is the effect on profit of a 10 per cent variation in estimation of each trade payab
> Report and Accounts of Craigielaw plc 1. What are the individual current asset items, by amount and as a percentage of total current assets, in the financial statements? 2. What is the effect on profit of a 10 per cent variation in the estimation of trad
> 1. What is the significance of the amount of property, plant and equipment in relation to the total assets of the company and to the business in general? Complete the table and write a short comment. 2. What is the relative significance of each type of a
> 1. List page numbers and a brief description of the evidence in the Report and Accounts that indicates that part of the document has been the subject of regulation. 2. What differences do you see between the audited and the unaudited parts of the Report
> Suggest reasons for the existence of more than one source of regulation of financial reporting by listed companies.
> Report and Accounts of Craigielaw plc 1. Read the paragraph headed ‘Directors’ Responsibilities’. What are the similarities and differences? 2. Read the ‘Report of the Auditors’ on the annual report. Compare this with the text book example. What are the
> Buchan Enterprises is considering investing in a new machine at a cost of £50,000. The respective cash flows are as follows: Year 1: 18,000 Year 2: 16,000 Year 3: 40,000 Year 4: 45,000 Year 5: 35,000 The company’s cost of capital is 12 per cent. Req
> In a calculation of the internal rate of return of a project it is found that the net present value is +£60m at 24% discount rate and −£20m at 26% discount rate. What is the Internal Rate of Return?
> Design a balanced scorecard for a restaurant business which owns three restaurants in the same town. Include three goals and three measurements of performance for each of the four aspects of the balanced scorecard, and write a short note justifying your
> Suggest three financial and three non-financial performance measures for a business which provides training in the workplace for updating word processing and computing skills. Each training course lasts two weeks and there is a standard fee charged per t
> Newcastle Patterns Ltd manufactures a single product that has the following standard cost specifications (per unit). During June, the following actual data have been recorded in the production of 1,400 units: Direct materials: 22,000 square metres at &A
> DEF Products Ltd manufactures and assembles one type of furniture unit. The following information is available for the year ended 31 August Year 7. The budgeted costs and the actual costs incurred during the year were as follows: The budgeted and actual
> (a) Define the term ‘budget’. (b) Explain how the budgetary process contributes to effective management in the areas of planning, control, communication and co-ordination and performance evaluation. (c) Motor Spares
> Kerryblue Ltd is a company manufacturing two products using one type of material and one grade of labour. Shown below is an extract from the company’s working papers for the next period’s budget. There are 12 5-day wee
> Sefton Ltd manufactures a product known as EC. The following information relates to the preparation of the budget for the year to 31 March Year 9: 1. Sales budget details for product EC are as follows: Expected selling price per unit: £100
> Explain why it is essential to have a statement of financial position (balance sheet), income statement (profit and loss account) and statement of cash flows in the annual report of a listed company.
> Smith plans to sell a single product at £10 each. His expected volume of sales for the first 2 months in business from 1 January to 28 February Year 1 is as follows: He sells his product for both cash and credit terms and estimates that 20 p
> Prepare: a production plan in units and (b) a purchases budget in £s for Houseline Ltd in Year 5, based on the following information that has been agreed by the line managers of Houseline Ltd as a basis for preparation of budgets for Year
> Farthing Ltd manufactures bicycle frames, which it sells for £40. The present machinery has a maximum capacity of 20,000 frames per year at which level net profit is £3 per frame and the contribution to sales ratio is 20%. A replacement machine is to be
> A valve-refurbishing business wishes to compare the cost of buying a component outside the business with the cost of making the component itself. The cost of outside components is £6 per item. The cost of manufacturing internally is calcul
> A business manufactures two products, namely chairs and tables, in a fixed ratio of four chairs to every one table. Sales and production volumes are always equal. The following selling prices and costs are specified: Total fixed costs of production are
