Institutt for fysikk, NTNU TFY4155/FY1003: Elektrisitet og magnetisme Spring 2005

Summary, week 17 (April 26 and 27)

Magnetic susceptibility and permeability [FGT 31.1; YF 28.8; TM 27.5, AF 26.7; LHL 26.1; DJG 6.4.1]

If the magnetization is proportional to the external field, we may write

M = χ_{m}H

Here, χ_{m} is magnetic susceptibility. Then, using the relation between B, H, and M:

B = μ_{0}(H + M) = μ_{0}(1 + χ_{m})H = μ_{0}μ_{r}H = μH

Here μ_{r} = 1+χ_{m} is relative permeability while μ is the permeability of the medium. (Cf. linear response in dielectric media!)

Some typical numbers: Diamagnets: χ_{m} ∼ −10^{−5} to −10^{−4} Paramagnets: χ_{m} ∼ 10^{−4} to 10^{−3} Ferromagnets: χ_{m} ∼ 10^{3} to 10^{4}

This means that only ferromagnetic materials (Fe, Co, Ni...) ”respond” significantly to an external magnetic field.

Electromagnetic induction [FGT 30.1 - 30.6; YF 29.1 - 29.5; TM 28.2; AF 27.1 - 27.3; LHL 24.1; DJG 7.2]

An electromotive ”force” (emf, actually a voltage) E is induced in a conducting wire loop if the magnetic flux ϕ_{m} that is enclosed by the loop varies with time:

E = −\frac{ϕ_{m}}{dt}

Enclosed magnetic flux is given by the surface integral of the magnetic field B, where the integral is taken over the surface S that is enclosed by the conducting loop:

ϕ_{m} = ∫_{S} B · dA

So we see that ϕ_{m} may vary with time in different ways, e.g. with

• time dependent enclosed area S

• time dependent orientation of the conducting loop (determined by the direction of dA)

• time dependent magnetic field B (direction and/or magnitude)

In all these cases we obtain an induced emf in the conducting loop.

The direction of E is determined by Lenz’ law: The current I generated by E creates a magnetic field B_{I} and hence a magnetic flux ϕ_{I} = ∫_{S} B_{I} · dA that is directed opposite to the flux change dϕ_{m} that caused E.

An induced emf E in a closed loop implies an induced electric field E:

E = ∮ E · dl

Faraday’s law hence expresses a connection between the fields E and B:

∮_{c} E · dl = −\frac{d}{dt} ∫_{S} B · dA

where c is a closed loop that encloses a surface S.

Since the integral of such a ”Faraday induced” electric field around a closed loop is not zero, it is, by definition not a conservative field. (Whereas an electrostatic field is conservative.)

Mutual inductance [FGT 32.1; YF 30.1; AF 27.12; LHL 25.4; DJG 7.2.3]

A current I_{1} in a current loop (1) results in a magnetic field B_{1} in the region around it. This field, we may, at least in principle, calculate via Biot-Savarts law:

B_{1} = \frac{μ_{0}I_{1}}{4π} ∮_{(1)} \frac{dl_{1} × r̂}{r^{2}}

(see e.g. week13.pdf). If another current loop (2) is located in this region, the magnetic field from loop (1) will result in a magnetic flux through loop (2):

ϕ_{2} = ∫_{S_{2}} B_{1} · dA_{2} = ∫_{S_{2}} \{\frac{μ_{0}I_{1}}{4π} ∮_{(1)} \frac{dl_{1} × r̂}{r^{2}}\} · dA_{2}

Whatever this integral may be, we may write

ϕ_{2} = M_{21}I_{1}

assuming I_{1} is constant everywhere in loop (1). (And it must be, otherwise charge would pile up somewhere...!) The factor M_{21} is the mutual inductance between the two loops (1) and (2) and expresses how much magnetic flux we get ”through” (or ”inside”) loop (2) when a current runs in loop (1):

M_{21} = \frac{ϕ_{2}}{I_{1}}

And the other way around, we obtain a magnetic flux inside loop (1) if a current runs in loop (2):

ϕ_{1} = ∫_{S_{1}} B_{2} · dA_{1} = ∫_{S_{1}} \{\frac{μ_{0}I_{2}}{4π} ∮_{(2)} \frac{dl_{2} × r̂}{r^{2}}\} · dA_{1}

i.e.: The current I_{2} in loop (2) creates the magnetic field B_{2}, and hence the flux ϕ_{1} through loop (1). And whatever this integral may be, we can always write

ϕ_{1} = M_{12}I_{2}

where the factor M_{12} expresses how much magnetic flux we get through loop (1) when a current runs in loop (2):

M_{12} = \frac{ϕ_{1}}{I_{2}}

Both M_{21} and M_{12} are simply geometric factors, depending on the shape, size, and relative positions of the two loops. One can show that

M_{21} = M_{12}

is always true. So, you may choose between two alternative ways of finding the mutual inductance between two current loops: Either find the magnetic flux through (1) due to the current in (2), or the other way around. Sometimes one of the alternatives is much easier than the other.

Unit for inductance: [M] = [ϕ_{m}/I] = [B · A/I] = T·m^{2}/A ≡ H (henry)

Mutual induction: A time dependent current I_{1}(t) in loop (1) results in a time dependent flux ϕ_{2}(t) through loop (2), and hence an induced emf in loop (2):

E_{2} = −\frac{dϕ_{2}}{dt} = −M_{21} \frac{dI_{1}}{dt}

A time dependent current I_{2}(t) in loop (2) results in a time dependent flux ϕ_{1}(t) through loop (1), and hence an induced emf in loop (1):

E_{1} = −\frac{dϕ_{1}}{dt} = −M_{12} \frac{dI_{2}}{dt}