Why is dB coming out of the page?

Because of the cross product between *I*dl and u_{r} . They are right angles to each other, so dB is at a right angle to both of them.

What is the equation to describe the magnetic flux in the center of a circular current loop?

What is Gauss's law for magnetic flux?

Total magnetic flux leaving a closed surface is zero

Ampère’s Law for Time-Invariant Currents in a Vacuum

State Faraday's law of induction

"The electromotive force (voltage V_{emf} ) induced in a closed circuit equals the negative rate of change of magnetic flux(Φ) linking that circuit"

How does the voltage change in a time-varying electric and magnetic field?

What is the significance of this?

From the equation, we can see that the voltage between points A and B depends not only on the voltage difference between the two but also the route taken from point A to B or vice versa.

The induced voltage is dependent on the path.

The significance of this is that voltage in a time-varying electric and magnetic field depends on the path between the points.

What direction will the wire move if a current is applied __out__ of the board?

How can the direction of the force be represented mathematically?

Using the right hand rule, the force is up.

Unit vector I crossed with unit vector B, gives F.

The value obtained with the cross product is orthogonal to the two other vectors, which is why the cross product is suitable.

How was this equation formulated?

Formulated from the observation that the force on a charge through a magnetic field with velocity **v** is perpendicular to v.

The magnitude of the force due to the magnetic field is proportional to the speed of the particle as well as the charge. Doubling either of those values doubles the force, they are linearly proportional.

From this experiment, the equation can be rationalised.

The equation is also sign sensitive on all the values.

Why can't the Lorentz force on a charge change the velocity?

According to lorentz's force, the force due to a magnetic field on a charge is always perpendicular to the velocity. It is only capable of changing the direction! It will never change the velocity as the direction of the force will always change to stay perpendicular to the velocity.

What equation would you use to calculate the force on a wire, if the length of the wire, current and magnetic flux density are known?

F = ILB

or

F = BIL

State the Biot-savart law outlined by lewin

Explain why the equation is expressed with these variables.

From experimentation it was found that B was proportional to I and as well as the distance from the wire squared.

Since it is proportional, they need to be multiplied by a constant.

This can be seen in the equation as CI/r^{2} .

The vectors dl and r-hat in the brackets represent the direction of B. The cross product is taken of both these vectors as B is perpendicular to dl and r.

Since r-hat is a unit vector, the purpose of this cross product is to only give the direction of the magnetic field.

What are the different forms of amperes law?

J_{f} is the free current density only

J is total current density

∮_{c} is line integral around the closed curve C

∬_{S} denotes a 2D surface integral over S enclosed by C

The curve C bounds both a surface S which the electric current passes through and encloses the current.

In other words, the law is a relation between the total amount of magnetic field around some path (line integral) due to the current that passes through the enclosed path (surface integral).