Advanced E&M Phys 350
where f is the force/charge. (Around the loop the electrostatic force integrates to zero).
There are various types of forces that we consider.
For case 2 we worked out in class how there is work done by the component of the magnetic field along the wire. This work is always equal and opposite the work done by the perpendicular component. The total work done by B remains zero. We found that this work was added by an external force and transferred to the charges through the electric and magnetic forces.
For case three we recognize that the electric field can be separated into two parts based on the way it is produced.
Notice that the total field (E+G from above) satisfies the correct equations for the total field. In circuits where we typically describe voltages and EMFs across elements (where the B field is changing), we are making this separation.
An understanding of EMF is important.
Because a current in a loop of wire produces a field and therefore an associated magnetic flux, there will be and EMF (type 3).
The sign is important. First you choose a direction of positive current flow. This is arbitrary (your choice). If I is increasing in this direction, then the EMF is negative. This means that the force/charge generated in the coil due to the increasing field opposes the flow of current in the positive direction. Any charge that moves through the circuit in this direction therefore looses energy. Current will only flow if there is an equal force pushing the charge across the EMF. A voltage difference must exist that produces this force. Work is done by the electrostatic force (due to D V) this energy is extracted by the coil which stores the energy in the magnetic field.
Thus for each component that develops an EMF there is a potential difference (D V) across the component such that the total force on a charge is zero and the steady flow of current is guaranteed. For resistors one could develop the notion of an average frictional force and equivalently consider the potential difference due to the balancing of frictional and electrostatic forces so that charge can move with an average constant velocity. For capacitors the electrostatic repulsion of the accumulated charge matches the electrostatic force of the potential difference applied (D V). Here the applied potential pushes the charges together and the energy goes into the electric field between the plates. There is no EMF with capacitors because the fields involved are both electrostatic.
Other important topics:
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