A charge q of 10.0 C is placed somewhere in space. What is the work required to bring a charge of 1.0 mC from a point X, 10.0 m from q, to a point Y, 2.0 m from q? Does the answer depend on which path the charge follows?

An electron is brought from infinity to a distance of 10.0 cm from a charge of −10.0 C. How much work was done on the electron?

$f(x)$ is continuous on $(-\infty, \infty)$. Use the given information to sketch the graph of $f$.

$$f(-2)=-1,f(0)=0,f(2)=1;$$

$$f'(-2)=0,f'(2)=0;$$

$$f'(x)>0\ \text{on}\ (-\infty,-2),(-2,2)\ \text{and}\ (2, \infty)$$

Simplify each expression.

$$\left( \frac { 8 } { 27 } \right) ^ { \frac { 2 } { 3 } }$$

Four equal charges of

$$5.0 \mu \mathrm { C }$$

are placed at the vertices of a square of side 10 cm. a. Calculate the value of the electric potential at the centre of the square. b. Determine the electric field at the centre of the square. c. How do you reconcile your answers to a and b with the fact that the electric field is the derivative of the potential?

Differentiate the given function. $y=x^{2}\left(x^{3}-6 x+7\right)$ [Hint: Multiply first.]

Two conducting spheres are separated by a distance that is large compared with their radii. The first sphere has a radius of 10.0 cm and has a charge of

$$2.00 \mu \mathrm { C }$$

on its surface. The second sphere has a radius of 15.0 cm and is neutral. The spheres are then connected by a long conducting wire. a. Find the charge on each sphere. b. Calculate the charge density on each sphere (charge density is the total charge on the sphere divided by the surface area of the sphere). c. Calculate the electric field on the surface of each sphere. d. Comment on your result in the light of your answer to part b. Why is it stated that the wire is long?

A uniformly charged rod of length L and total charge Q lies along the x axis. (a) Find the components of the electric field at the point P on the y axis a distance d from the origin. (b) What are the approximate values of the field components when d>>L? Explain why you would expect these results.

a. Determine the electric potential at the mid point of the line joining two equal positive charges q in terms of q, the Coulomb constant and the charge separation d. b. Repeat a for two equal but opposite charges.

a. If $\int_{1}^{3} f(x) d x=2 \text { and } \int_{1}^{6} f(x) d x=-3, \text { find } \int_{3}^{6} f(x) d x .$ b. If $\int_{0}^{2} f(x) d x=5, \int_{4}^{6} f(x) d x=-2 \text { and } \int_{0}^{6} f(x) d x=7, \text { find } \int_{2}^{4} f(x) d x$

A 40.0-kg child swings in a swing supported by two chains, each 3.00 m long. The tension in each chain at the lowest point is 350 N. Find the force exerted by the seat on the child at the lowest point. (Ignore the mass of the seat.)

Explain the difference between filtrate and urine.

Consider two particles of mass m and 16m separated by a distance d. a. Deduce that at point P, a distance

$$\frac { d } { 5 }$$

from the particle with mass m, the gravitational field strength is zero. b. Determine the value of the gravitational potential at P.

Find a general solution, using the method of undetermined coefficients for a particular solution of the nonhomogeneous equation. $y^{\prime \prime}-6 y^{\prime}+8 y=3 e^{x}$